Nonlinear Waves in Nonlinear Waves in Granular Crystals Granular - - PowerPoint PPT Presentation
Nonlinear Waves in Nonlinear Waves in Granular Crystals Granular Crystals Mason A. Porter Oxford Centre for Industrial and Applied Mathematics Mathematical Institute, University of Oxford References: PRE 77 : 015601(R) (2008); Physica D 238 : 6,
Oxford Centre for Industrial and Applied Mathematics Mathematical Institute, University of Oxford
References: PRE 77: 015601(R) (2008); Physica D 238: 6, 666 (2009); arXiv:0802.1451 (to appear in Mech.
Theorists
Ricardo Carretero-González (San Diego State University)
Fernando Fraternali (University of Salerno)
Panos Kevrekidis, Georgios Theocharis (University of Mass. at Amherst)
Yi Ming Lai (Oxford)
Laurent Ponson (Caltech)
Experimentalists
Nick Boechler, Chiara Daraio, Devvrath Kahtri, Ivan Szelengowicz (Caltech)
Eric Herbold (GTRI)
Georgios Theocharis ???
Granular crystals = tightly-packed
arrangements of beads
Compressive force when they squeeze
each other (no force when not in contact)
Use elastic waves to study
phenomena such as those associated with disorder or with electromagnetic waves
Analogy with photonic crystals
band gaps in the presence of precompression
(“phononic crystals”)
Can achieve either weakly nonlinear
Hertzian contact ⇒ 3/2 interaction power Ej = elastic modulus of bead j (how stiff is it) νj = Poisson ratio (how much squeezing is caused by stretching) mj = mass Rj = radius yj = coordinate of center of bead j
Without gravity and
precompression, the equations of motion cannot be linearized (one just gets 0). We observe waves with finite support.
Periodic array of cells. Propagation of acoustic waves through
inhomogeneous but periodic media
Each cell contains N1 beads of one type
and N2 beads of a second type.
Consider particles of different m, E, ν
Examine interactions between nonlinearity + material heterogeneity (e.g., periodic arrangements)
What happens to pulse width, propagation speed, etc?
Left: Steel:PTFE Right: Steel:rubber Top: experiments Bottom: numerics
Left: 2:1 dimers Right: 5:1 dimers Top: experiments Bottom: numerics
Like the FPU → KdV calculation, but more intricate Generalize asymptotic analysis of homogeneous chains
by Nesterenko using a generalized version of method by Pnevmatikos, Flytzanis, & Remoissenet (PRB 33, 4, 2308 [1986]).
For 1:1 chains only Two different types of beads give the following rescaled
equations (general interaction power k):
Compacton solutions of long-wavelength
Traveling waves: u = u(ξ) = u(x - Vst)
The solitary pulse in the experiments consists of one arch
Key experimentally testable properties:
(Peak) force-velocity scaling:
Hertzian interactions: Observation: Same value for homogeneous, N1:N2 chains, and trimer
chains (independent of mass ratio). This is a fundamental property of the Hertzian interactions and hence arises from the geometry of the beads. (We only have an analytical calculation for monomers and 1:1 dimers.)
Pulse width = π/β
Depends on mass ratio but independent of pulse’s amplitude Fundamentally different for dimer versus homogeneous chains
Closed form expression for β (too long to write down)
Example: stainless steel:PTFE 1:1
dimer chain
Top: force-velocity scaling
Experiments versus numerics (purple curve)
Green curve is numerics with a different value of E for PTFE
Bottom: pulse width (full width
at half maximum) versus particle number
Red curve: numerics
Green curve: experiments
Line 1: theoretical (homogenized) width for m1 >> m2
Line 2: theoretical width for correct masses (ratio about 4:1)
Line 3: theoretical width for m1 = m2
Steel:bronze:PTFE (1:1:1)
configurations with different values of m, E, and ν
Physica D, 2009
F. Fraternali, MAP, &
CD, arXiv:0802.1451 (to appear in Mech. Adv.
Genetic algorithm allows
control over the selection
Example: Given fixed
particle sizes, particle materials, and chain length, arrange them to minimize the maximum amplitude of the transmitted impulse.
0906.4094
Homogeneous chain except for a small number of lower-mass particles
Initial condition: actuator with sinusoidal oscillation, which we then turn off (numerical).
Chain is precompressed
1 defect, 2 nearest-neighbor defects, 2 next-nearest-neighbor defects
Breathers in dimer chains (theory + numerics + experiment; in preparation)
MAP, PGK, & CD, PRL 102: 024102
Carefully match experimental characterization of loss rates with numerical experiments
Posit a power-law form of dissipation and fit to try to determine the best exponent and coefficient
Obtain consistent value for different materials
Conclusion: the exponent is decidedly different from 1 (i.e., the linear dashpot case) previously used in the literature to add dissipation to these models
N S
Solitary wave propagation in periodic chains
Chains of dimers and chains of trimers
Good agreement in force-velocity scalings and pulse width (numerics + theory + experiments)
Other heterogeneous configurations
Chains with defects: impurity modes
“Magnetization” transition in disordered chains
“Optimized” chains to obtain desired properties (minimize peak force at the end)
Incorporation of dissipation
More to come…
More on optical modes/gap solitons/defects, quasiperiodic and randomized arrangements of beads, systematic incorporation of plastic and viscoelastic effects, other interaction exponents (including nonuniform interactions)
General theme: Nonlinearity + Material Inhomogeneity = Fun!
Applications too: Non-invasive defect detection, shock absorbers, etc.
A newsstand magazine article on the FPU