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M Metamaterials i l with tunable dynamic properties Varvara Kouznetsova Marc Geers 6 October 2015 Mechanics of Materials Project aim development of new generation mechanical metamaterials h i l i l with adaptive, tunable or superb

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SLIDE 1

M i l Metamaterials with tunable dynamic properties

Varvara Kouznetsova Marc Geers Mechanics of Materials 6 October 2015

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SLIDE 2

Project aim

development of new generation h i l i l mechanical metamaterials with adaptive, tunable or superb dynamical properties by systematically exploiting the combination of material and geometrical non-linearities g potential applications:

tunable wave guides tunable wave guides adaptive passive vibration control superdumping p p g acoustic diodes acoustic cloaking

/ mechanics of materials

noise insulation

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SLIDE 3

Outline

  • Background
  • Locally resonant metamaterials
  • State of the art and challenges
  • Towards addressing the challenges
  • Plan of work
  • International collaborations

/ mechanics of materials

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SLIDE 4

Background: wave propagation

  • Wave - disturbance or oscillation that travels through

matter or space accompanied by a transfer of energy matter or space, accompanied by a transfer of energy without mass transfer

  • Electromagnetic waves
  • Electromagnetic waves
  • do not require medium
  • Mechanical waves
  • Mechanical waves
  • propagate by local deformation of a medium

→ dynamic properties of materials → dynamic properties of materials

/ mechanics of materials

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SLIDE 5

Background: dispersion properties

  • infinite homogeneous material
  • infinite periodic material

l

wave length

x

l

Bragg scattering, phononic crystal (PC)

g

ncy

band gap

  • infinite material with local resonators

frequen

band gap

wave number

x

l

g p

locally resonant acoustic metamaterials (LRAM)

frequency: number of ‘oscillations’ per second wave number: number of ‘oscillations’ over specified distance wave number: 1/wave length

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SLIDE 6

Background: working principle of LRAM

incident wave

/ mechanics of materials

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SLIDE 7

Potential applications of LRAM

  • low frequency absorbers
  • noise reduction

noise reduction

[Zhao et al. J. App. Phys. (2010)]

  • negative refractive index w.r.t. sound waves
  • super lenses

super lenses

  • cloaking

[Zhu et al. Nature Comm. (2014)]

  • exotic dynamic effective properties
  • fluid like behaviour
  • fluid-like behaviour

(zero shear stiffness)

  • compressive and
  • compressive and

shear wave filters

[Lai et al. Nature Mat. (2011)]

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SLIDE 8

Example of LRAM

epoxy rubber [Liu, Z., et al. Science (2000)] lead

Frequency band gaps Frequency band gaps

lattice constant =15.5mm band gap freq 380 Hz > band gap freq. 380 Hz ->

  • approx. 300x lattice const.
  • coating material ?
  • core material?

/ mechanics of materials

core material?

  • volume fraction?
  • size variations?
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SLIDE 9

Coating properties

  • vol. frac. = 40%
  • Rin = 5 mm
  • vol. frac. = 40%
  • Rin = 5 mm
  • R

= 7 5 mm

  • Rex = 7.5 mm
  • Rex = 7.5 mm
  • coating Poisson’s ratio:  = 0.469

(longitudinal wave velocity cl=23 m/s)

  • coating Poisson’s ratio:  = 0.49998

(longitudinal wave velocity cl >1000 m/s)

  • (in)compressibility of coating changes the band gap structure

[Krushynska, Kouznetsova, Geers, JMPS (2014)]

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SLIDE 10

Inclusion volume fraction & core material

1st band gap 1st band gap

(inclusion and coating sizes fixed)

W

  • lowest bound is independent of volume fraction (local resonance)
  • band gap width depends on the volume fraction with a maximum around 70%
  • heavier inclusions result in lower and wider band gap

[Krushynska, Kouznetsova, Geers, JMPS (2014)]

  • tungsten (W) is a good option instead of lead
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SLIDE 11

Two inclusion sizes combined

  • same core radius
  • different coating

thickness

  • presence of different inclusion sizes increases the number of band gaps
  • presence of different inclusion sizes increases the number of band gaps
  • but the width of band gaps is decreased
  • due to the localized nature of in-plane modes, overlapping band cannot be created
  • dispersion properties can be fine-tuned for a specific application

[Krushynska, Kouznetsova, Geers, JMPS (2014)]

dispersion properties can be fine tuned for a specific application

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SLIDE 12

State of the art and Challenges

State of the art:

li l ti t i l

  • linear elastic materials
  • (mostly) infinite medium
  • r specific geometries only (e g spheres)
  • or specific geometries only (e.g. spheres)

Challenges: Challenges:

  • non-linear materials?
  • finite structures (i e real applications)?
  • finite structures (i.e. real applications)?

− boundaries/constraints? complex loading?

  • tunable dynamic behaviour?

tunable dynamic behaviour?

/ mechanics of materials

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SLIDE 13

State of the art and Challenges

State of the art:

li l ti t i l

  • linear elastic materials
  • (mostly) infinite medium
  • r specific geometries only (e g spheres)
  • or specific geometries only (e.g. spheres)

Challenges: Challenges:

  • non-linear materials?
  • finite structures (i e real applications)?
  • finite structures (i.e. real applications)?

− boundaries/constraints? complex loading?

  • tunable dynamic behaviour?

tunable dynamic behaviour?

/ mechanics of materials

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SLIDE 14

Computational Homogenization

momentum balance

homogenize

initial & bnd. conditions

strain velocity stress momentum strain momentum initial boundary value problem applicable to problem applicable to

  • finite structures
  • complex loading/constraints
  • non-linear material behaviour

[Pham, Kouznetsova, Geers, JMPS (2013)]

non linear material behaviour

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SLIDE 15

Computational homogenization: example

macro velocity profile

[Pham, Kouznetsova, Geers, JMPS (2013)]

macro v(t) macro micro velocity distribution 2

  • 2
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Closed-form Homogenization

momentum balance

homogenize

evolution eq. q closure relations

  • nly once for a given material
  • static-dynamic

y g static dynamic decomposition

  • model order

reduction dynamic microfluctuation field field applicable to

  • finite structures

[Sridhar, Kouznetsova, Geers, in preparation]

  • finite structures
  • complex loading/constraints
  • linear material behaviour
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SLIDE 17

Closed-form Homogenization: example

homogenized with homogenized with dynamic fluctuations homogenized without homogenized without dynamic fluctuations

[Sridhar, Kouznetsova, Geers, in preparation]

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SLIDE 18

State of the art and Challenges

State of the art:

li l ti t i l

  • linear elastic materials
  • (mostly) infinite medium
  • r specific geometries only (e g spheres)
  • or specific geometries only (e.g. spheres)

Challenges: Challenges:

  • non-linear materials?
  • finite structures (i e real applications)?
  • finite structures (i.e. real applications)?

− boundaries/constraints? complex loading?

  • tunable dynamic behaviour?

tunable dynamic behaviour?

/ mechanics of materials

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SLIDE 19

Other effects of non-linearities

  • material non-linearities lead to amplitude dependent

di i b h i dispersion behaviour

  • Prof. Michael Leamy and co-workers:

i t ith ‘ k’ li iti − spring-mass systems with ‘weak’ non-linearities

  • geometrical non-linearities can switch-on/off band gaps
  • geometrical non-linearities can switch-on/off band gaps
  • Prof. Katia Bertoldi and co-workers

different levels of applied compressive strain different levels of applied compressive strain

[Wang et al. PRL (2014)]

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SLIDE 20

Project aim

development of new generation h i l i l mechanical metamaterials with adaptive, tunable or superb dynamical properties by systematically exploiting the combination of material and geometrical non-linearities g potential applications:

tunable wave guides tunable wave guides adaptive passive vibration control superdumping p p g acoustic diodes acoustic cloaking

/ mechanics of materials

noise insulation

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SLIDE 21

Project plan

  • focus on development of analysis and modelling techniques

for non linear metamaterials for non-linear metamaterials

  • combination of techniques from non-linear vibrations (e.g.

harmonic balance, perturbation method etc.) with transient , p ) computational homogenization

  • LRAMs with continuous phases and realistic non-linear

t i l ti material properties

− non-linear rubber elasticity − visco-elasticty visco-elasticty − visco-plasticity − ….

  • LRAMs with geometrically non-linear effects
  • identify the most critical material and geometrical properties

for tunable systems

  • formulate design guidelines
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International Collaboration

  • Prof. Michael Leamy (Georgia Institute of

Technology USA) Technology, USA)

  • non-linear phenomena in dynamics and

metamaterials metamaterials

  • Prof. Katia Bertoldi (Harvard University, USA)
  • geometrically non-linear effects in metamaterials

geometrically non linear effects in metamaterials

  • Prof. John Willis (University of Cambridge, UK)

( y g , )

  • mathematical aspects of dynamics of metamaterials
  • Prof. Norman Fleck (University of Cambridge, UK)
  • design, manufacturing and testing of structured

g , g g materials

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M i l Metamaterials with tunable dynamic properties

Varvara Kouznetsova Marc Geers Mechanics of Materials 6 October 2015