Andrew Norris, norris@rutgers.edu
Static and dynamic elastic homogenization of periodic structures
The talk considers (i) static homogenization of lattice structures modeled by thin beam members and (ii) a procedure for dynamic homogenization of general periodic elastic media. The seemingly disparate topics are motivated by a need to understand the static and dynamic behavior of lattice structures which have been proposed as
candidates for pentamode materials. Our main result for static properties of a lattice
Pstructure with coordination
number Z is that the effective moduli can be expressed in Kelvin-like form C = N
i,j=1 Pij Ui ⊗ Uj where
N = 12 Z(Z + 1), Ui are second order tensors, and Pij are elements of a N × N projection matrix P of rank
N − d, in d = 2 or 3 dimensions. The N second order tensors {Ui } split into Z stretch dominated and N − Z
bending dominated elements. The latter contribute little to the stiffness in the limit of very thin members, in
which case the elastic stiffness is, at most, of rank Z − d. C is rank one if Z = d + 1, corresponding to
pentamode materials.
Part (ii) describes a general procedure for defining and calculating dynamic effective properties of periodic
media. The dynamic homogenization yields constitutive relations for the effective medium of the so-called
Willis type which couples stress with velocity, and momentum to strain. It turns out that there is then a unique
system of equations governing the effective field variables defined according to heff (x, t) = hhi ei(k·x−ωt)
for each field variable (displacement, stress, etc.) where hhi is the spatial average of the periodic part of h. This
homogenization scheme clearly reduces to the static procedure for ω = 0, k = 0, while the effective dynamic
parameters (stiffness, density, etc.) which are functions of ω and k, have the important property that they yield
the exact dispersion relation for Bloch waves. Implications of the dynamic homogenization scheme for lattice
networks will be discussed and compared with the static effective moduli.
Static and dynamic elastic homogenization
of periodic structures
Andrew N. Norris
Rutgers University
Continuum Models Discrete Systems
Salt Lake City, Utah, July 21-25 2014
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
Kelvin (1856)
Positive definite strain energy :
Kelvin (1856)
Positive definite strain energy :
Necessary and sufficient conditions for truss to be rigid
Maxwell (1864)
Condition for rigidity of frameworks is
Z > 5 in 2D
Z > 11 in 3D respectively
Deshpande et al. JMPS (2001).
Kelvin (1856)
Positive definite strain energy :
Necessary and sufficient conditions for truss to be rigid
Necessary and sufficient condition for rigidity of 2D and 3D
frameworks is Z > 5 and Z > 11, respectively
Deshpande et al. JMPS (2001)
Maxwell(1864)
Bell (1907)
Octet truss
1903
Octet truss
1903
Octet truss
2014
1903
Ultra-light, Ultra-stiff Mechanical
Metamaterials MIT/LLNL
science.1252291
stiffest lattice structure
Gurtner & Du a d, “tiffest elastic et o ks , P‘“A
4 doi: 10.1098/rspa.2013.0611
soft modes
2D lattices in shear
soft/easy
modes
Gurtner & Durand, PRSA 2014 doi: 10.1098/rspa.2013.0611
soft modes are bending dominated
i.e. effective static moduli, are stretch dominated
soft/easy
modes
Gurtner & Durand, PRSA 2014 doi: 10.1098/rspa.2013.0611
Kelvin
Milton and Cherkaev (1995)
Unimode
Bimode
……
…….
Pentamode (PM)
PM: five zero eigen-stiffnesses
diamond-like structure
five easy/soft modes
pentamode lattice structures
Kadic et al., APL 2012
Mejica and Lantada Smart Mat. Struct. 2013
Kadic et al.,
NJP 2013
Schnitty et al., APL 2013
mechanical behavior of pentamode materials (PM)
a single type of stress (and strain)
- like hydrostatic stress and volumetric strain of a liquid
g
“microstructure”
static equilibrium
... under gravity
PM = li iti g ase of a isot opi solids ith ze o shea
igidit
water as a pentamode material
elastodynamics
acoustics
transformation acoustics:
isotropic PM
Norris (2008, 2009)
pentamode form of stiffness:
anisotropic PM
Metal Water
generic structure for transformation acoustics in water
Norris, Nagy (2011)
islands
struts
- for stiffness mainly
- inertial role mainly
cell size
bulk modulus = 2.25 GPa
density
= 1000 kg/m^3
shear modulus = 0.065 Gpa
(i.e. small)
transparent to sound in water
Pentamode material and transformation acoustics
same amount of total empty (cloaked) space
heavy metal preferred
oids i isi le
volume of empty space remains constant
conservation of empty/cloaked space = conservation of mass
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
lattice unit cells
3 ways to deform members
3 ways to deform members
Stretching
Beam bending
Node bending
Strain energy
3 ways to deform members
Z = coordination number
Stretching
Beam bending
Node bending
compliances
Strain energy
(compliance = inverse of stiffness)
Ingredients:
Z = coordination #
length, direction
V = unit cell volume
axial compliance
bending compliance
node compliance
Ingredients:
Z = coordination #
length, direction
V = unit cell volume
axial compliance
bending compliance
node compliance
Effective elastic moduli
Elasti moduli of lattice networks, from stiffest to pentamode ANN (submitted)
equilibrium
stress
strain
equilibrium
stress
strain
affine deformation
macro deformation
(strain, rotation)
nonlinear
local rotation
local displacement
equilibrium
stress
strain
affine deformation
macro deformation
nonlinear
local rotation
linearize
local displacement
solve for
equilibrium
stress
strain
affine deformation
macro deformation
local rotation
local displacement
Ingredients:
Z = coordination #
length, direction
V = unit cell volume
axial compliance
bending compliance
node compliance
Effective elastic moduli
Elasti moduli of lattice networks, from stiffest to pentamode ANN (submitted)
bulk modulus
diamond
FCC
R
R
R
R
BCC
octet truss
a
E = You g s
odulus
5 lattices
(all cubic symmetry)
bulk modulus:
diamond
FCC
E = You g s
shear moduli:
R
R
R
R
BCC
octet truss
a
tetrakaidecahedral
-
can be made isotropic
14 = 6+8
odulus
Ingredients:
Z = coordination #
length, direction
V = unit cell volume
axial compliance
bending compliance
node compliance
Effective elastic moduli
stretch
P is a projector
flex
node
stretch dominated effective elastic moduli
Z=14, d=3
fully stiff
Z > 11, d=3
Z > 5, d=2
pentamodal
Z=d+1
Z=6, d=2
Z=4, d=3
Z=3, d=2
stretch dominated lattices
Z = coordination #
length, direction
axial compliance
V = unit cell volume
effective elastic moduli
P is a projector
stretch dominated lattices
Z = coordination #
length, direction
axial compliance
V = unit cell volume
effective elastic moduli
P is a projector
pentamode: Z=d+1
Explicit moduli for PM lattices (Z=d+1)
Z=3, d=2
2D PM lattices (all isotropic)
pentamode:
Z=4, d=3
length, direction,
axial compliance,
V = cell volume
PM lattices
+ 5 more, small
PM lattices
+ 5 more, small
Poisso s atio
m
n
depends entirely on the 5 small eigenvalues because
PM lattices
+ 5 more, small
Poisso s atio
m
n
depends entirely on the 5 small eigenvalues because
e.g.
PM lattices
Poisso s ratio depends only on the geometry
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
Example: cubic lattice
Lattice dynamics: for each rod
node i
node j
1) Longitudinal wave equation
2) Flexural wave equation
bending in orthogonal directions
longitudinal
node i
node j
Total force at point i from rod ij:
Equilibrium:
Floquet conditions
= set of nodes connected to node i
dispersion relation
bending
Honeycomb: 2D pentamode
beam theory
Aluminum beams
length : thickness = 12.5 : 1
COMSOL
torsional mode
beam theory
Aluminum beams
length : thickness = 20 : 1
COMSOL
Diamond lattice : 3D pentamode
steel rods
beam theory
length : thickness = 20 : 1
COMSOL
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
« METALindex
WATER
» STRUCTURE
metal water as negative
material
(NIM)
discovery of Anne-Christine Hladky-Hennion (IEMN / Lille)
0.04
sonic
lines
FREQUENCY (MHz)
0.03
X

0.02
0.01
Aluminium structure
0.00
X
J
G
X
negative group velocity at sonic speed
NIM with matched
- index
(speed)
- impedance (transmission) - not quite!
density of water
equi-frequency contour
circle
e.g. NIM acoustic lens
S
I
Metal water as negative index material
image
Source
source
F = 80 kHz
Image
APL, 102, 144103, 2013
50
S
I
focusing with MW flat lens
lens with 7 rows
simulation
source
imperfect focusing related to effective impedance at finite frequency
- not well understood
need for dynamic effective medium theory
0.04
sonic
lines
- has density of water
- sonic phase speed
FREQUENCY (MHz)
0.03
0.02
0.01
0.00
X
G
J
X
impedance mismatch at finite frequency for some wave numbers reduces focusing potential
spectral curves + static properties
not enough to describe finite frequency effects
need a dynamic effective medium theory (what does it even mean?)
e.g. f e ue cy depe de t effective i peda ce
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
exact dynamic effective medium theory
periodically layered, 2D & 3D periodic media
Static: effective moduli well defined
Dynamic: what do we mean by
d a i effe ti e ediu ?
exact dynamic effective medium theory
periodically layered, 2D & 3D periodic media
Static: effective moduli well defined
Dynamic: what do we mean by
d a i effe ti e ediu ?
no small parameter
exact dynamic effective medium theory
periodically layered, 2D & 3D periodic media
Static: effective moduli well defined
Dynamic: what do we mean by
d a i effe ti e ediu ?
consider fixed wavenumber and frequency
acoustic equations
/ SH wave motion
velocity
potential
dilatation
canonical
form
divergence
form
SH:
d is
o e tu
v is st ess
φ is “H displa e e t
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
1D - periodically layered medium
Monodromy matrix:
= matricant (propagator) over unit period
1D - periodically layered medium
Monodromy matrix:
uniform medium:
= matricant (propagator) over unit period
1D - periodically layered medium
Monodromy matrix:
uniform medium:
= matricant (propagator) over unit period
One definition: dynamic homogenization of layered medium
x
(effective parameters)
•
cannot be identified as the system matrix for elasticity.
•
uniquely defines a material with Willis constitutive behavior.
(= generalized elasticity that couples stress with momentum)
• Homogenized parameters depend on the start point (y=0) in the unit cell.
In that sense it is not a unique dynamic effective theory.
• Effective medium gives exact results for reflection/transmission from half-space y>0
• 1D only
ANN, A. L. Shuvalov, A. A. Kutsenko, and O. Poncelet, Effective Willis constitutive equations for
periodically stratified anisotropic elastic media, Proc. R. Soc. A, 467, 1749-1769 (2011)
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
Another definition of dynamic effective medium equations
Bloch wave / Floquet theory :
h = pressure, displacement, etc.
h(x) periodic with average
over
dynamic effective medium equations
Bloch wave / Floquet theory :
h = pressure, displacement, etc.
h(x) is periodic with average
over
dynamic homogenization:
for any ω and k find the equations governing
• Natural generalisation of static homogenisation
• 1D, 2D, 3D
• Unique effective material
method
• write equations in divergence form with general forcing on RHS
• use Plane Wave Expansion (Fourier series) for clarity/simplicity
dynamic effective medium equations
Problem: find the equations governing
• effective equations are of Willis form
• Bloch wave dispersion ωn = ωn(k) relation drops out
Results:
• ω, k → gi es sta da d stati ho oge izatio
• Unique dynamic effective medium
• 1D, 2D, 3D periodic
ANN, A. L. Shuvalov and A. A. Kutsenko, Analytical formulation of 3D dynamic homogenization
for periodic elastic systems, Proc. R. Soc. A, 468 1629-165 (2012)
acoustic equations in canonical form
velocity
potential
dilatation
canonical divergence form:
Results in a nutshell:
originally
periodic
Effective equations are of Willis form:
matrix
is Hermitian,
all are functions of
material parameters independent of x
is real,
(but not of x)
is complex
Bloch waves
effective equations:
= Bloch wave equation
dynamic effective medium equations: elasticity
momentum balance:
stress and momentum in terms of
strain and particle velocity
Willis equations
Explicit expressions for effective properties using plane wave expansion (PWE) .
E.g. effective density is anisotropic
Norris et al. (Proc. R. Soc. A 2012)
d a i G ee s
at i i PWE
dynamic effective medium equations: elasticity
closure of Willis eqs under DEMT
inhomogeneous Willis material
+ dynamic homogenization
effective Willis material
Willis equations
Explicit expressions for effective properties using plane wave expansion (PWE) .
E.g. effective density is anisotropic
Norris et al. (Proc. R. Soc. A 2012)
d a i G ee s
at i i PWE
dynamic effective medium equations: elasticity
closure of Willis eqs under DEMT
inhomogeneous Willis material
+ dynamic homogenization
effective Willis material
like transformation acoustics
(Milton, Briane, Willis 2006)
Willis equations
Explicit expressions for effective properties in terms of (infinite) vectors/matrices
using plane wave expansion (PWE) . E.g. effective density is anisotropic
Norris et al. (Proc. R. Soc. A 2012)
d a i G ee s
at i i PWE
Dynamic effective medium : 1D example
(for shear waves: density
, shear stiffness
)
example: 1D periodic medium
layer
thickness
density
stiffness
1
0.37
1
1
2
0.313
2
7
3
0.317
0.5
0.33
effective properties vs. wavenumber
Floquet branches
second branch
first branch
(fundamental)
static values
0
1
Dynamic effective medium: 1D example
Bloch wave phase seed:
reflection / transmission
Floquet branches
uniform medium
effective medium
where
is complex
properties on 2nd branch
even if
cannot impedance match
with a uniform medium
• Discrete lattices
- static effective properties
-- stiffest to pentamode
- dynamics
• Continuous periodic media
- dynamic effective medium theories
-- 1D theory
-- 1,2,3D theory
Summary
static effective moduli of periodic lattices:
• includes stretch, flex and node bending
•stretch dominated drops out
• explicit pentamode
dynamic effective medium theories for periodic systems:
• 1D based homogenisation special case
•
•
•
•
use <h(x) > gives natural extension of static homogenization
rigorous at all frequencies, wavenumbers
effective equations of Willis type, parameters computable by PWE
framework for solving BVPs, e.g. scattering/reflection
Summary
static effective moduli of periodic lattices
•explicit pentamode
dynamic effective medium theories for periodic systems
• using <h(x) > gives natural extension of static homogenization
-- at all frequencies, wavenumbers
• effective equations of Willis type, parameters computable by PWE
• framework for solving BVPs, e.g. scattering/reflection
Need to broaden how we think of periodic materials dynamically
thanks:
A. Nagy, X. Su
A. Shuvalov, A. Kutsenko
A.C. Hladky-Hennion
J. Cipolla, N. Gokhale
Rutgers
U. Bordeaux 1
IEMN Lille
Weidlinger
ONR , NSF, U. Bordeaux, Fulbright
& thanks to
CMDS 13 organizers
for the hard work
& you
for listening!