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, 666 (2009); arXiv: 0802.1451 (to appear in Mech. Adv. Mat. Struct. ), PRL 102 : 024102 (2009); arXiv: 0904.0426 ; arXiv: 0906.4094

Coauthors Coauthors  Theorists Ricardo Carretero-González (San Diego State University)  Fernando Fraternali (University of Salerno)  M. Kavousanakis, I. G. Kevrekidis (Princeton)  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 ??? 

Fermi-Pasta-Ulam (FPU) problem Fermi-Pasta-Ulam (FPU) problem

Granular Crystals Granular Crystals  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 or strongly nonlinear dynamics

Equations of Motion (1D) Equations of Motion (1D)  Hertzian contact ⇒ 3/2 interaction power  E j = elastic modulus of bead j (how stiff is it)  ν j = Poisson ratio (how much squeezing is caused by stretching)  m j = mass  R j = radius  y j = coordinate of center of bead j

Highly nonlinear solitary waves Highly nonlinear solitary waves  Without gravity and precompression, the equations of motion cannot be linearized (one just gets 0). We observe waves with finite support.

Chains of Dimers Chains of Dimers  Periodic array of cells.  Propagation of acoustic waves through inhomogeneous but periodic media  Each cell contains N 1 beads of one type and N 2 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? 

1:1 dimer chains 1:1 dimer chains Left: Steel:PTFE Right: Steel:rubber Top: experiments Bottom: numerics

N 1 :1 ste 1 steel:PT PTFE d E dimer c r chains s Left: 2:1 dimers Right: 5:1 dimers Top: experiments Bottom: numerics

Long wavelength asymptotics Long wavelength asymptotics  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 Compacton Solutions  Compacton solutions of long-wavelength PDE  Traveling waves: u = u( ξ ) = u(x - V s t)

Compacton Solutions II Compacton Solutions II  The solitary pulse in the experiments consists of one arch of the cosine profile.  Key experimentally testable properties:  (Peak) force-velocity scaling:  Hertzian interactions:  Observation: Same value for homogeneous, N 1 :N 2 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 Obtain both known limiting cases (m 1 = m 2 , m 1 >> m 2 )   Closed form expression for β (too long to write down)

Theory/numerics vs. Experiments Theory/numerics vs. Experiments  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 m 1 >> m 2 Line 2: theoretical width for correct  masses (ratio about 4:1) Line 3: theoretical width for m 1 = m 2 

Chains of Trimers Chains of Trimers • Examine configurations with different values of m, E, and ν • See MAP et al, Physica D , 2009 Steel:bronze:PTFE (1:1:1)

Optimization Optimization  F. Fraternali, MAP, & CD, arXiv: 0802.1451 (to appear in Mech. Adv. Mat. Struct.)  Genetic algorithm allows control over the selection of multiple parameters  Example: Given fixed particle sizes, particle materials, and chain length, arrange them to minimize the maximum amplitude of the transmitted impulse.

Optimized Decorated/Tapered Chain Optimized Decorated/Tapered Chain

Breathers in Chains with Defects Breathers in Chains with Defects G. Theocharis, M. Kavousanakis, PGK, CD, MAP, & I. G. Kevrekidis, arXiv:  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)

Breathers: 2 defects Breathers: 2 defects

Incorp rporat ating d g dissipat ation n R. Carretero-González, D. Kahtri,  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

Elast stic sp c spin c n chains s • L. Ponson, N. Boechler, Y. M. Lai, MAP, PGK, & CD, arXiv:0904.0426 N S • Order parameter: “Magnetization” = | # [NS] – # [SN] | / (total # of pairs)

Conclusions Conclusions  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. 

And now for something completely And now for something completely different… different…  A newsstand magazine article on the FPU problem: MAP, N.J. Zabusky, B. Hu, & D. K. Campbell, American Scientist 97 , 6, 214 (2009)