Describing metamaterials with generalized continuum models...

Describing metamaterials with generalizedcontinuum models: the relaxed micromorphic

approach

Marco Valerio d’Agostino

joint work with:

Angela Madeo (Lyon), Patrizio Neff (Essen), Gabriele Barbagallo (Lyon),Ionel-Dumitrel Ghiba (Iasi), Bernhard Eidel (Siegen), Domenico Tallarico (Lyon),

Alexis Aivaliotis (Lyon).

5th-9th November, MECAWAVE, Villa Clythia, France

1 / 47 Marco Valerio d’Agostino Relaxed micromorphic approach

Motivation

wave propagation in micro-structuredsolids,

non linear dispersion curves, acousticand optic branches, band-gaps,direction dependent (anisotropic),

goal: describe ”high” frequencyresponse of multi-scale materialswithout explicitly resolving themicro-structure geometry,

but taking into account thesymmetries of the micro-structure (viaNeumann’s principle).


Neumann’s principle: how symmetries of the material enterthe constitutive law

Attributed by Voigt (disciple of Neumann),

PRINCIPLE: assumed constitutive law describes thephysics of the system. Every lattice symmetry of thecrystal must also be a symmetry of the assumedconstitutive law.

In many cases lattice symmetry is equal to symmetry ofassumed constitutive law.

Exception: linear elasticity used to describe hexagonal latticesymmetry, assumed constitutive law will be isotropic.Hexagonal symmetry is subset of isotropy.

Our assumed anisotropic constitutive law will be based onNeumann’s principle for each constitutive tensor.


Typical high frequency response - direction dependent


State of the art


Our goal


Outline

Recapitulation of the relaxed micromorphic model.

Wave propagation in anisotropic relaxed micromorphicmedium, comparison with dispersion curves for givenmicrostructure based on Bloch-Floquet analysis.

Decisive tool: recently discovered micro-macrohomogenization formula exclusive to anisotropic relaxedmicromorphic medium.

Decisive tool: numerical homogenization on the unit-celllevel based on periodic and Dirichlet boundary conditions.

Reduction of PDE-system to an algebraic system, withplane-wave ansatz.

Comparison of the dispersion relations and polar plots ofphase velocity with results from Bloch-Floquet analysis.


The relaxed micromorphic model

The kinematics of the relaxed micromorphic model is described by two fields:

u : Ω× [0,T ]→ R3, P : Ω× [0,T ]→ R3×3,

where

u is the macroscopicdisplacement,

P is the micro-distortiontensor field,

Ω is the referenceconfiguration in R3.


The classical micromorphic model (Eringen, Mindlin)

The structure of the energy of the classical micromorphic model is

E(∇u,P,∇P) =1

2

⟨C (∇u − P),∇u − P

⟩+

1

2

⟨C(∇u − P), symP

⟩+

1

2〈Cmicro symP, symP〉+

µ L2c

2〈∇P,∇P〉

whereC, C : R3×3 → R3×3 is dimensionless 4thorder elasticity tensor,

Cmicro : Sym (3)→ Sym (3) is dimensionless 4thorder elasticity tensor,

Lc characteristic length scale.


Potential energy density

W (∇u,P,CurlP) =1

2〈Ce sym (∇u − P) , sym (∇u − P)〉︸︷︷︸

anisotropic elastic - energy

+1

2〈Cmicro symP, symP〉︸︷︷︸micro - self - energy

+1

2〈Cc skew (∇u − P) , skew (∇u − P)〉︸︷︷︸

invariant local anisotropic rotational elastic coupling(related to Cosserat couple modulus µc )

+µ L2

c

2〈CurlP,CurlP〉︸︷︷︸curvature

whereCe ,Cmicro,L : Sym (3)→ Sym (3) classical 4thorder elasticity tensors,

Cc : so (3)→ so (3) 4th order coupling tensors (6 independent components),



Comparison between the two energies


Kinetic energy density

J (u,t ,∇u,t ,P,t) =1

2〈ρ u,t , u,t〉+

1

2〈Jmicro symP,t , symP,t〉+

1

2〈Jc skewP,t , skewP,t〉

+1

2〈T sym∇u,t , sym∇u,t〉+

1

2〈Tc skew∇u,t , skew∇u,t〉

+µ L2

c

2〈CurlP,t ,CurlP,t〉

where

ρ : Ω→ R+ macro-inertia mass density,

Jmicro : Sym (3)→ Sym (3) classical 4thorder tensor (free micro-inertia density),

T : Sym (3)→ Sym (3) classical 4thorder tensor (gradient micro-inertia density),

Jc ,Tc : so (3)→ so (3) 4thorder coupling tensors (6 independent components),




By suitable restriction we obtain many particular theories of theclassical micromorphic model:

the linear isotropic microvoids model [Cowin-Nunziato (1983)theory] in dislocation format: P = ϑ(x1, x2, x3) · 1, ϑ ∈ R;

the microstrain model [Forest-Sievert (2006)]: P ∈ Sym (3);

the linear isotropic asymmetric microstretch (Eringen’smicrostretch theory) model in dislocation format;

the linear isotropic Cosserat model: P ∈ so (3);

the Popov-Kroner (2001) dislocation model;

the symmetric earthquake structure model of Teisseyre (1973);

the gauge theory of dislocations of Lazar and Anastassiadis(2008).



S. Owczarek, I.D. Ghiba, M.V. d’Agostino P. Neff. Non standard micro-inertia terms in the relaxedmicromorphic model: well-posedness for dynamics. arXiv:1809.04791


First variation and balance equations

The first variation of the action functional

δ

∫I

∫Ω

(J (u,t ,∇u,t ,P,t)−W (∇u,P,CurlP)) dv dt.

yields the Euler-Lagrange equations (3 coupled blocks foru, symP and skewP):

ρ u,tt −Div (T sym∇u,tt)−Div (Tc skew∇u,tt) = Div (Ce sym (∇u − P) + Cc skew (∇u − P)) ,

Jmicro symP,tt = Ce sym (∇u − P)− Cmicro symP − µ L2c symCurlCurlP,

Jc skewP,tt = Cc skew (∇u − P)− µ L2c skewCurl CurlP.

whose solutions characterize the equilibrium points of the system.


Why CurlP in the relaxed micromorphic model?

Simplified statics∫Ω

(µe ‖sym (∇u − P)‖2 + µc ‖skew (∇u − P)‖2 + µmicro ‖symP‖2 + L2

c ‖CurlP‖2)dx

Penalty µe , µc →∞ ⇒ P = ∇u. Limit energy∫

Ωµmicro ‖sym∇u‖2 dx

≈ linear elasticity with Cmicro

Response of the unit cell: Lc →∞ ⇒ CurlP = 0, P = ∇ϑ, ∇ϑ = ∇u.Limit energy

∫Ωµmicro ‖sym∇u‖2 dx ≈ linear elasticity with Cmicro

Classical micromorphic elasticity: penalty µe , µc →∞⇒ second

gradient elasticity∫

Ω

(µmicro ‖sym∇u‖2 + L2

c

∥∥D2u∥∥2)dx

Classical micromorphic elasticity: Lc →∞ infinite stiffness of the unitcell: only homogeneous deformations remain possible

Classical micromorphic elasticity: homogeneous microstructure∇P = 0⇒ P = const.

Relaxed micromorphic elasticity: fluctuating microstructure is possible

CurlP = 0⇒ P = ∇ϑ16 / 47 Marco Valerio d’Agostino Relaxed micromorphic approach

Well-posedness of the model: static and dynamics

The dynamical problem can be written as a variational problem in a suitableBanach space: considering the spaces of functions H1

0(Ω) and H0(Curl; Ω),where

H0(Curl; Ω) := P ∈ L2(Ω,R3×3) | curlPi ∈ L2(Ω,R3), i = 1, 2, 3

and P·τ = 0, ∀τ tangent to ∂Ω,

a weak solution of the given system of PDEs is a function

(u,P) ∈ C 2(

[0,T ] ; H10(Ω)× H0 (Curl; Ω)

)verifying the variational problem

W1 ((u,tt(t),P,tt(t)), (ϕ,Φ)) =W2 ((u(t),P(t)), (ϕ,Φ)) + l (f (t),M(t))(ϕ,Φ).

for all (ϕ,Φ) and for all t ∈ [0,T ], where (f ,M) ∈ H−1(Ω)× H0(Curl,Ω)∗ and

l (f (t),M(t))(ϕ,Φ) := 〈f , ϕ〉H−1,H10

+ 〈M,Φ〉H0(Curl)∗,H0(Curl)


Well-posedness of the model: static and dynamics

W1 ((u,P), (ϕ,Φ)) :=

∫Ω

(ρ 〈u, ϕ〉+ 〈Jmicro symP, symΦ〉+ 〈Jc skewP, skewΦ〉

+ 〈T sym∇u, sym∇ϕ〉+ 〈Tc skew∇u, skew∇ϕ〉

+ µ L 2c 〈CurlP,CurlΦ〉

)dx

W2 ((u,P), (ϕ,Φ)) :=

∫Ω

(〈Ce sym(∇u − P), sym(∇ϕ− Φ)〉

+ 〈Cc skew(∇u − P), skew(∇ϕ− Φ)〉

+ 〈Cmicro symP, symΦ〉+ µ L 2c 〈CurlP,CurlΦ〉

)dx

With a fixed-point theorem argument, it can be proven that

Theorem

For every (f ,M) ∈ C 0([0,T ] ; H−1(Ω)× H0(Curl; Ω)∗

),

(u0,P0) , (u0,P0) ∈ H10(Ω)× H0(Curl; Ω) and for all tensors Ce ,Cmicro,Cc and

Jmicro, Jc ,T,Tc which satisfy the given conditions, there exists a unique solutionof the induced variational problem.


Korn-type inequality for incompatible tensor fields

Neff et al. (2011-2016) proved that there exists a constant C(Ω) > 0 such that

∀ P ∈ H(Curl; Ω,R3×3), P × n|∂Ω = 0

we have the two fowwing estimates:

‖P‖L2(Ω) ≤ C(Ω)(‖symP‖L2(Ω) + ‖CurlP‖L2(Ω)

),

and‖P‖L2(Ω) ≤ C(Ω)

(‖dev symP‖L2(Ω) + ‖dev CurlP‖L2(Ω)

),

If P = ∇u, then it turns into a version of Korn’s inequality

∀ u ∈ H1(Ω,R3), ∇u × n|∂Ω = 0 : ‖∇u‖L2(Ω) ≤ C(Ω) ‖sym∇u‖L2(Ω) .

S. Bauer, P. Neff, D. Pauly and G. Starke. Dev-Div-and DevSym-devCurl-inequalities for incompatiblesquare square tensor fields with mixed boundary conditions. ESAIM Control Optim. Calc. Var.,22(1):112-133, 2016.

P. Neff, D. Pauly, K.J. Witsch. Poincare meets Korn via Maxwell: Extending Korn’s first inequality toincompatible tensor fields. J. Diff. Equations. 258(4):1267-1302, 2014.


Plane wave propagation

Monochromatic plane wave ansatz

u (x , t) = u e i (〈k,x〉−ωt), P = P e i (〈k,x〉−ωt), k = k k ,

u = (u1, u2, u3) is the polarization vector in R3,

k = (k1, k2, k3) ∈ R3,∥∥k∥∥ = 1 is the direction of wave

propagation,

k = ‖k‖ is the wave number and ω is the frequency,

P =(P11, P12, P13, P21, P22, P23, P31, P32, P33

)∈ R3×3.

The PDE system reduces to:

D (k , ω) · v = 0,

where D (k, ω) is a 12× 12 matrix and v is the vector of amplitudesgiven by the components of u and P. Calculating

det D (k , ω) = 0,

we find the dispersion curves ω = ω(k).


Necessary and sufficient conditions for real wavepropagation in the isotropic case

Sylvester’s criterion states that a Hermitian matrix M ispositive-definite if and only if the leading principal minors arepositive. Hence, it is possible to prove the following proposition:

The dynamic relaxed micromorphic model admits real wavevelocity ω

k if and only if

µc ≥ 0, µe > 0, 2µe + λe > 0,

µmicro > 0, 2µmicro + λmicro > 0,

(µmacro > 0), 2µmacro + λmacro > 0,

κe + κmicro > 0, 4µmacro + 3κe > 0.

The requirement µmacro > 0 is redundant, since it is implied byµe , µmicro > 0 and the Cosserat couple modulus µc only needsto be non-negative for real wave velocity ω

k .


Necessary and sufficient conditions for real wavepropagation

The necessary and sufficient conditions for real wave propagationare already implied by the smaller set:

µe > 0, µmicro > 0, µc ≥ 0, κe + κmicro > 0,

2µmacro + λmacro > 0.

It is easy to notice that:

positive definiteness of the energy =⇒ real wave propagation

P. Neff, A. Madeo, G. Barbagallo, M. V. d’Agostino, R. Abreu, I.-D. Ghiba, (2017) Real wave propagationin the isotropic-relaxed micromorphic model. Proceedings of the Royal Society A: Mathematical,Physical and Engineering Sciences. DOI: 10.1098/rspa.2016.0790


Legendre-Hadamard (strong) ellipticity conditions

Classical elasticity: W = W (∇u)

∀ ξ, η ∈ R3\0 : t → W (∇u + t ξ ⊗ η)

is convex in t at all ∇u.Legendre-Hadamard =⇒ real wave propagation

strong ellipticity

Micromorphic elasticity: W = W (∇u,P,∇P)

Considering z = (u,P) ∈ R12, ∇z ∈ R36 ' R12 ⊗ R3:

∀ ξ ∈ R3\0, η ∈ R12\0 : t → W (z ,∇z + t ξ ⊗ η)

is convex in t at all (z ,∇z).

Legendre-Hadamard 6=⇒ real wave propagationstrong ellipticity


Strong ellipticity for the relaxed micromorphic model(isotropic case)

The relaxed micromorphic energy W (∇u,P,CurlP) is:

W =µe ‖sym (∇u − P)‖2 +λe2tr (∇u − P)2 + µc ‖skew (∇u − P)‖2

+ µmicro ‖symP‖2 +λmicro

2(tr (P))2 +

µ L2c

2‖CurlP‖2 .

The strong ellipticity condition for the relaxed micromorphicmodel is:

strong ellipticity ⇐⇒ µe + µc > 0 , 2µe + λe > 0 .

This condition is not sufficient to imply real wave propagation:

strong ellipticity 6=⇒ real wave propagation


Unit-cell and Bloch-Floquet analysis

The metamaterial Ω is aperiodic medium withtetragonal symmetric unit cell.

Grey region of Σc is filled byaluminum while the white oneis empty, ρ is 1485 kg/m3

Dispersion curves via theBloch-Floquet analysis for wavepropagation directions

k = (1, 0, 0)

k = (√

2/2,√

2/2, 0).

a b c d E ν

1 0.9 0.3 1 70 0.33

mm mm mm m GPa −


Tetragonal case

The group of the tetragonal symmetry is the Dihedral group D4:

QaiQbjQchQdk Cabcd = Cijhk , ∀Q ∈ D4.

The matrices corresponding to the considered tensors (Voigt notation) are

Ce =

2µe + λe λe λ∗e 0 0 0λe 2µe + λe λ∗e 0 0 0

λ∗e λ∗e (Ce)33 0 0 0

0 0 0 (Ce)44 0 0

0 0 0 0 (Ce)44 00 0 0 0 0 µ∗e

, Cc =

4µ∗c 0 00 4µ∗c 00 0 4µc

,

(same structure for Cmicro, L, Jmicro, T), (same for Lc , Jc , Tc ).


Tetragonal case

Wave propagation in the plane (x1, x2, 0): further reduction to

Ce =

2µe + λe λe ∗ 0 0 0

λe 2µe + λe ∗ 0 0 0∗ ∗ ∗ 0 0 00 0 0 ∗ 0 00 0 0 0 ∗ 00 0 0 0 0 µ∗e

, Cc =

∗ 0 00 ∗ 00 0 4µc

,

Cosserat couple modulus µc

The remaining parameters of the tetragonal relaxed micromorphic model are

(µe , λe , µ∗e , µc , µmicro, λmicro, µ

∗micro, Lc) and (ρ, η1, η2, η3, η

∗1 , η1, η2, η3, η

∗1 )

and we set the characteristic length scales Lc = Lc = 0.


Micro-macro homogenization formula (tensor harmonicmean)

For Lc → 0 the relaxed micromorphic continuum approximates a classical macroscopicCauchy medium whose elasticity tensor Cmacro is related to Cmicro and Ce of therelaxed micromorphic model via the ”Reuss-like” homogenization formula (harmonictensor mean)

Cmacro = Cmicro (Cmicro + Ce)−1 Ce ⇐⇒

Ce = Cmicro (Cmicro − Cmacro)−1 Cmacro.

Indeed, starting from the system

Div [Ce sym(∇u − P) + Cc skew(∇u − P)] = 0, (1)

Ce sym(∇u − P) + Cc skew(∇u − P)− Cmicro symP − µ L2c (CurlCurlP) = 0,

in the limit Lc → 0+, we obtain

Ce sym(∇u − P) + Cc skew(∇u − P)− Cmicro symP = 0

whose symmetric part gives the two following equations

Ce sym(∇u − P) = Cmicro symP (2)

and

Ce sym∇u = (Cmicro + Ce) symP ⇔ symP = (Cmicro + Ce)−1Ce sym∇u (3)



Replacing (2) and (3) in (1), we obtain

Div [Cmicro symP] = 0 ⇔ Div[Cmicro (Cmicro + Ce)−1 Cesym∇u

]= 0.

This formula is impossible in classical micromorphic elasticity (needsspecific split in sym and skew).

In the tetragonal case:

µmacro =µe µmicro

µe + µmicro, µ∗

macro =µ∗e µ

∗micro

µ∗e + µ∗

micro

,

2µmacro + 3λmacro =(2µe + 3λe) (2µmicro + 3λmicro)

2 (µe + µmicro) + 3 (λe + λmicro).

Barbagallo G., Madeo A., d’Agostino M.V., Abreu R., Ghiba I.D., and Neff P. Transparent anisotropy forthe relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics.International Journal of Solids and Structures,Vol. 120, 7-30, 2017.



Limit for large, engineering-size samples −→ linear elasticity withCmacro

Limit for unit-cell structure −→ linear elasticity with Cmicro

Both Cmicro and Cmacro are available:

Cmacro ←− periodic homogenization or Bloch-Floquet (long wave-lengthresponse, tangents of the acoustic curves, infinite periodic medium)

Cmicro ←− calculation of the stiffness of the unit-cell

Cmicro

relaxed micromorphic↓

interpolate−−−−−−−−−−−−−−−−−−−−−−−−−−−−→

taking into accountthe actual specimen

size

Cmacro


Identification of Cmacro - periodic homogenization

Periodic boundary conditions mimic an infinite number of unit cells.

Calculation of macroscopic stiffness Cmacro by numerical homogenization.

λmacro µmacro µ∗macro

1.738 5.895 0.620

[GPa] [GPa] [GPa]

1

2〈Cmacro E , E〉 |Ωc | :=

inf

∫ξ∈Ωc

1

2

⟨C (ξ) sym

(∇ξv (ξ) + E

), sym

(∇ξv (ξ) + E

)⟩dξ∣∣∣ v ∈ C∞(Ωc ,R3

)is periodic

,

where C(ξ) is the elasticity tensor of the aluminum phase and air depending on

the position of ξ in the unit-cell and E = sym∇u (x).

Periodic stiffness Cmacro is independent of the size and shape of unit-cell.


Identification of Cmicro - stiffest tetragonal unit-cellunder affine Dirichlet boundary conditions (KUBC)

Calculation of microscopic stiffness Cmicro by numerical homogenization.Energy minimization problem∫

Ω

W (∇u,P,CurlP) dm −→ min (u,P) ∈ H1(Ω)× H(Curl; Ω) ,

whit boundary conditions

u|∂Ω (x) = B · x , B ∈ R3×3, ∇u|∂Ω (x) · τ1,2 = P|∂Ω (x) · τ1,2,

τ1,2 are tangent vectors to ∂Ω. The minimization problem

inf(u,P)

∫x∈Ω

W (∇u,P,CurlP) dx

can be bounded above by taking P (x) = ∇u (x). Indeed

inf(u,P)

∫x∈Ω

W (∇u,P,CurlP) dx ≤ infu

∫x∈Ω

W (∇u,∇u, 0) dx

= infu

∫x∈Ω

1

2〈Cmicro sym∇u (x) , sym∇u (x)〉 dx .



Therefore, the maximal possible stored elastic energy of the relaxedmicromorphic model over an arbitrary window Ω ⊂ Ω is

infu

∫x∈Ω

1

2〈Cmicro sym∇u (x) , sym∇u (x)〉 dx , u|∂Ω (x) = B · x ,

Taking Ω as the unit cell V (x)

1

|V (x)|

∫ξ∈V(x)

∇ξu (x + ξ) dξ =1

|V (x)|

∫ξ∈V(x)

B dξ = B,

and symmetrizing

ε =1

|V |

∫ξ∈V (x)

ε (x + ξ) dξ = symB = E .

From convexity for homogeneous boundary conditions v(ξ) = B · ξ

inf

∫ξ∈V (x)

1

2〈Cmicro sym∇ξv (ξ) , sym∇ξv (ξ)〉 dξ

∣∣∣ v : V (x)→ R3, v |∂V (x) (ξ) = B · ξ

=1

2

⟨Cmicro symB, symB

⟩|V (x)| =

1

2

⟨Cmicro E ,E

⟩|V (x)| .



We demand that the fine-scale energy should equal the coarse-scale energy over

the same domain V (x) , under the same affine boundary conditions B · ξ and

for the same material. This implies that for all B ∈ R3×3:

1

2

⟨Cmicro E , E

⟩|V (x)|︸︷︷︸

coarse scale micromorphicupper energy limit

≥ (4)

inf

∫ξ∈V (x)

1

2

⟨C (ξ)

(sym∇ξv (ξ) + E

), sym∇ξv (ξ) + E

⟩dξ∣∣∣ v ∈ C∞0 (

V (x) ,R3)

︸︷︷︸fine-scale linear elastic energy

.

According to the Hill-Mandel energy equivalence, we can define a uniqueapparent stiffness tensor CV

KUBC, independent of E ,

inf

∫ξ∈V (x)

1

2

⟨C (ξ)

(sym∇ξv (ξ) + E

), sym∇ξv (ξ) + E

⟩dξ∣∣∣ v ∈ C∞0 (

V (x) ,R3)

︸︷︷︸fine-scale linear elastic energy

=1

2|V (x)|

⟨CVKUBC E , E

⟩.

(5)



Extended Neumann’s principle allows us to determine the tensor Cmicro

by requiring

∀E ∈ Sym (3) :⟨Cmicro E ,E

⟩≥⟨CV

KUBC E ,E⟩, (6)

and ⟨C∗micro E ,E

⟩≥⟨Cmicro E ,E

⟩for any other possible tensor C∗micro which verifies (6).

Dirichlet-stiffness Cmicro depends on the size and shape of unit cell.

In general, bigger unit cell less stiff.

Cmicro is defined unambiguously (Lowner matrix supremum). Indeed,

for all (x , y , z)T ∈ R3

⟨2µ + λ λ 0

λ 2µ + λ 00 0 µ∗

Cmicro

xyz

,x

yz

⟩ ≥ ⟨2µ + λ λ 0

λ 2µ + λ 00 0 µ∗

CV

KUBC

xyz

,x

yz

⟩ ,(7)

where (x , y , z) represents(E 11,E 22,E 12

).



Many possible unit-cells: size, shape, symmetry,...

λhom µhom µ∗hom

(a) 4.37 6.242 8.332

(b) 2.125 5.899 2.264

(c) 5.270 8.927 4.042

(d) 5.981 6.254 4.96

Cmicro

[GPa] [GPa] [GPa]

Neumann’s principle: restriction to tetragonal unit-cells



Picture by M. Collet, Ecole central Lyon.



Denoting the four KUBC-entries by µi , λi , µ∗i , i = 1, 2, 3, 4, respectively, the question

is how to obtain the values µ, λ, µ∗ such that⟨2µ+ λ λ 0

λ 2µ+ λ 00 0 µ∗

Cmicro

xyz

,

xyz

⟩ ≥ ⟨2µi + λi λi 0

λi 2µi + λi 00 0 µ∗i

xyz

,

xyz

⟩ ,i = 1, 2, 3, 4, is always verified. Since µ∗ sits on the diagonal, we must necessarilyhave that

µ∗ ≥ µ∗i ∀ i = 1, 2, 3, 4.

Therefore, the problem is reduced to the 2× 2 block: for all (x , y)T ∈ R2⟨(2µ+ λ λ

λ 2µ+ λ

)(xy

),

(xy

)⟩≥⟨(

2µi + λi λiλi 2µi + λi

)(xy

),

(xy

)⟩, i = 1, 2, 3, 4.

According to the Sylvester-criterion for the difference of positive definite tensors, wemust have µ ≥ µi and µ+ λ ≥ µi + λi for i = 1, 2, 3, 4. We choose

µ := maxiµi , µ∗ := max

iµ∗i , µ+ λ := max

iµi + λi , λ := max

iµi + λi − µ.

This determines µ, µ∗, λ uniquely.


Fitting procedure

Cut-off frequencies for

det D(k, ω) = 0 at k = 0

are

ωr =

√µcη2, ωs =

√µe + µmicro

η1,

ω∗s =

√µ∗e + µ∗micro

η∗1, ωp =

√µe + µmicro + λe + λmicro

η1 + η3.


Dispersion and anisotropy in tetragonal metamaterials


Phase velocity ω(k)k : polar plots

Phase velocity ω(k)k

: polar plots as a function of the direction of wave

propagation k . The plotted curves have been calculated for three different

values of the wave number k.42 / 47 Marco Valerio d’Agostino Relaxed micromorphic approach

Phase velocity ω(k)k : polar plots

Phase velocity ω(k)k

: polar plots as a function of the direction of wave

propagation k . The plotted curves have been calculated for three different

values of the wave number k.43 / 47 Marco Valerio d’Agostino Relaxed micromorphic approach

Impulses


Conclusions

Our framework allows to describe anisotropic dispersion inmetamaterials while avoiding unnecessary complexificationsrelated to the introduction of elastic tensors of order higher thanfour, as it is the case for higher gradient elasticity (exception:hexagonal symmetry).

The model describes dispersion and anisotropy, not only for thefirst two acoustic modes, but also for other modes at higherfrequencies.

Complete a priori identification of static material parameters bynumerical homogenization: Cmacro (periodic bc on any unit-cell),Cmicro (affine Dirichlet on stiffest unit-cell).

Perspective: modeling metamaterials in a simplified framework, soproviding a concrete possibility for the design of complexmetastructures.

Improved calibration for high wave numbers (small wavelength)based on new non local inertia term CurlP,t will be investigated infuture works.


Our relaxed team

Thank you for your attention!

Msc. Alexios Aivaliotis,INSA Lyon, France

Dr. Gabriele BarbagalloINSA Lyon, France

Assoc.Prof. Marco V. d’AgostinoINSA Lyon, France

Prof. Bernhard EidelUniversitat Siegen, Germany

Lecturer I.-Dumitrel GhibaUniversity of Iasi, Romania

Prof. Angela MadeoINSA Lyon, France

Prof. Patrizio NeffUni. Duisburg-Essen, Germany

Dr. Domenico TallaricoINSA Lyon, France


References

d’Agostino M.V., Barbagallo G., Ghiba I.D., Eidel B., Neff P., Madeo A. Effiective description ofanisotropic wave dispersion in mechanical metamaterials via the relaxed micromorphic model. Submitted,https://arxiv.org.

P. Neff, I.-D. Ghiba, A. Madeo, L. Placidi, and G. Rosi. A unifying perspective: the relaxed linearmicromorphic continuum. Continuum Mechanics and Thermodynamics, 26(5):639-681, 2014.

Neff, P., Madeo, A., Barbagallo, G., d’Agostino, M. V., Abreu, R., Ghiba, I.-D. (2016). Real wavepropagation in the isotropic relaxed micromorphic model. Proceedings of the Royal Society A,473(2197):20160790, 2017.

Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G. (2015). Wave propagation in relaxed micromorphiccontinua: modeling metamaterials with frequency band-gaps. Continuum Mechanics andThermodynamics, 27(4–5), 551–570.

Madeo, A., Neff, P., Ghiba, I.-D., Rosi, G. (2016). Reflection and transmission of elastic waves in non-localband-gap metamaterials: a comprehensive study via the relaxed micromorphic model. Journal of theMechanics and Physics of Solids, 95, 441–479.

Madeo, A., Barbagallo, G., d’Agostino, M. V., Placidi, L., Neff, P. (2016). First evidence of non-locality inreal band-gap metamaterials: determining parameters in the relaxed micromorphic model. Proceedings ofthe Royal Society A, 472(2190), 20160169.

Madeo, A., Neff, P., d’Agostino, M. V., Barbagallo, G. (2016). Complete band gaps including non-localeffects occur only in the relaxed micromorphic model. Comptes Rendus Mecanique.http://doi.org/10.1016/j.crme.2016.07.002

d’Agostino M.V., Barbagallo G., Ghiba I.D., Madeo A., Neff P. (2017) A panorama of dispersion curves forthe weighted isotropic relaxed micromorphic model. Zeitschrift fur Angewandte Mathematik undMechanik. DOI: 10.1002/zamm.201600227


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