Mathematical justification of the apparition of the electromagnetic coupling

(1)

R E S E A R C H

Open Access

Mathematical justiﬁcation of the apparition

of the electromagnetic coupling

Somsak Orankitjaroen

1,3*

_{, Christian Licht}

1,2,3

_{and Thibaut Weller}

2

*_{Correspondence:} [email protected]

1_{Department of Mathematics,}

Faculty of Science, Mahidol University, Bangkok, Thailand

3_{Centre of Excellence in}

Mathematics, CHE, Bangkok, Thailand

Full list of author information is available at the end of the article

Abstract

By using the asymptotic analysis method as regards two-scale convergence we derive the eﬀective behavior of a ﬁne mixing of piezoelectric material and piezomagnetic material. It can be shown that an electromagnetic coupling arises only when each phase is connected and the interface is electrically and magnetically impermeable.

Keywords: Homogenization; Two-scale convergence; Piezoelectricity; Piezomagnetism

1 Introduction

Developing smart materials or multifunctional structures by using product properties of composite materials is a quite old idea but it, however, keeps getting more and more inter-esting for many technological applications. The concept is simple (see [1]): an effect in one of the phases of the composite implies a second effect in one of the other phases. Here we focus on the magnetoelectric coupling emerging in a composite material made of a mag-netostrictive phase and a piezoelectric phase. This kind of situation is interesting because the direct coupling between electric and magnetic fields is seldom found in nature and, when it exists, it is very weak. The composite materials made of ferroelectric and ferro-magnetic phases, however, may generate a considerably higher magnetoelectric response. This domain of research falls within the field of metamaterials which possess properties that cannot be found naturally. Understandably this opens the door for many different technological applications such as data storage, mechanical devices, magnetic sensors, high frequency signal treatment, etc. It is therefore of interest to propose an efficient and accurate modeling of the behavior of a composite material made of a piezoelectric phase and a piezomagnetic phase. This was done in [2] considering multilayered structures and using the asymptotic expansion method. It is worthwhile to refer to [3] and the references quoted therein to get a good overview of the problem. Here we propose a mathemati-cally rigorous study of this situation by using asymptotic analysis method such as two-scale convergence. We consider a periodically heterogeneous composite material made of a piezoelectric phase and a piezomagnetic phase. We introduce two other parameters that refer to the connectedness of each phase and to the boundary conditions on the interface. Depending on these two parameters, we show that different models appear when the size of the period goes to zero. The situations when a full coupling among mechanical, electric

(2)

and magnetic eﬀects (i.e., nonvanishing elasto–magneto–electric coupling coeﬃcients in the linearized context considered here) appears are highlighted.

2 Setting the problem

LetYea domain ofR3included inY:= (0, 1)3with a Lipschitz-continuous boundary. For all real positive numbersε, letEε

e:=

i∈Z3ε(i+Ye), which is assumed to be connected in R3_{, and}_Eε

m:=R3\E

ε

e,Ym:=Y\Ye. We will consider a body occupying a domain ofR3 with a Lipschitz-continuous boundary∂Ωmade of a purely piezoelectric phase occupying

Ω_eε:=Ω∩Eε

eand a purely piezomagnetic phase occupyingΩmε :=Ω∩Eεm. So the magne-toelectromechanical state of the body is described by the tripletsε_{:= (}_uε_,_φε_,_ψε_{) where}

uε_,_φε_,_ψε_{denote the ﬁeld of displacement, the electrical potential, the magnetic potential}

deﬁned inΩ,Ω_eε,Ω_mε, respectively. Ifσε,Dε_,_Bε_{denote the stress, the electric induction,}

the magnetic induction, respectively, one has

σε,Dε=Me

euε,∇φε inΩ_eε, σε,Bε=Mm

euε,∇ψε inΩ_mε, (1)

wheree(uε_{) is the strain associated with}_uε_while_M

eandMmstand for the piezoelectric and piezomagnetic tensors, respectively, withMe,MminLin(S3) satisfying

∃α> 0 s.t. Me(e,h)·(e,h)≥α(e,h)2,

Mm(e,h)·(e,h)≥α(e,h)2 ∀(e,h)∈S3×R3, (2)

whereLin(S3_{) denotes the space of linear mappings on}_S3_{the space of symmetric 3}_×₃ matrices whose scalar product and norm are denoted by·and| · |as inR3_.

We will consider various situations indexed byp= (p1,p2) in{1, 2} × {1, 2, 3, 4}. The case

p1= 1 corresponds toΩeεandΩmε connected,p1= 2 corresponds toΩeεconnected butΩmε disconnected. (Of course, by exchanging the words electric and magnetic the following results may be adapted if it is the magnetic phase only which is connected!) LetΓε

I :=

Ω∩∂Eε

e(=Ω∩∂Eεm) be the interface between the two phases, when

p2= 1: ΓIεis assumed to be electrically and magnetically impermeable (∂neφ

ε₌_∂ nmψ

ε₌

0) onΓε

I wherene and nm= –ne denote the normal outward toΩeε and Ωmε, respectively;

p2= 2: the electric and magnetic potentials are assumed to be constant on each con-nected component ofΓε

I ;

p2= 3: ΓIεis assumed to be electrically impermeable while the magnetic potential has to be constant on each connected component ofΓ_Iε;

p2= 4: the role played by electricity and magnetism in the previous case (p2= 3) is ex-changed.

Eventually, we assume that the electric and magnetic potentials take given valuesφ0and

ψ0onΓe,extε :=∂Ω∩E

ε

eandΓm,extε :=∂Ω∩E

ε

(3)

L2₍_Ω_;_R3₎_×_L2₍_Γ

N;R3), a weak formulation of the equilibrium problem for the body is

Pε

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Findsε_{= (}_uε_,_φε_,_ψε_{) in (0,}_φ

0,ψ0) +HΓ1D(Ω;R 3₎_×_Vε

e,p2×V

ε

m,p2s.t.

ΩeεMe(e(u

ε_),_∇_φε₎_·₍_e₍_v_),_∇_φ₎_dx₊

Ωmε Mm(e(u

ε_),_∇_ψε₎_·₍_e₍_v_),_∇_ψ₎_dx

=_Ωf·v dx+_Γ

Ng·v dH2 ∀(v,φ,ψ)∈H 1

ΓD(Ω;R 3₎_×_Vε

e,p2×V

ε

m,p2

with

V_e,1ε :=H_Γ1ε e,ext

Ω_eε,

Vε

m,1:=

ψ∈H_Γ1ε m,ext

Ω_mεwith vanishing average on each connected

component ofΩ_mε whose boundary does not meetΩ;

V_e,2ε :=H₀1Ω_eε,

V_m,2ε :=ψ∈H_Γ1ε m,ext

Ω_mεsuch that for all connected componentsΩ_mε,iofΩ_mε

there existsCε,i(ψ) inRwithψ:=ψ–Cε,i(ψ) inH₀1Ω_mε,i;

V_e,3ε :=V_e,1ε , V_m,3ε =V_m,2ε ;

Vε

e,4:=Ve,2ε , Vm,3ε =Vm,1ε , where for all open sets GofRn_, _H1

γ(G;Rn) denotes the subspace of the Sobolev space

H1₍_G_;_Rn_{) made of the elements with vanishing trace on}_γ _{included in the boundary}_∂_G

ofG.

By the Lax–Milgram lemma, (Pε_{) has a unique solution}_sε_{. The very question is to study}

the asymptotic behavior whenεgoes to zero which will supply the eﬀective behavior of the heterogeneous body.

3 A convergence result

To study the convergence whenεgoes to zero, we use the two-scale convergence method [4] and recall the deﬁnition.

Deﬁnition 1 A sequence of functionsvε_in_L2₍_Ω_;_R3_{) is said to two-scale converge to a}

limitv0_{belonging to}_L2₍_Ω_×_Y_;_R3_{) if for any function}_θ_in_D₍_Ω_;_C∞

# (Y;R3)), we have

lim

ε→0

Ω

vε(x)·θ(x,x/ε)dx=

Ω×Y

v0(x,y)·θ(x,y)dx dy,

whereD(Ω;C∞_# (Y;R3_{)) denotes the space of inﬁnitely smooth and compactly supported} functions inΩ with values in the spaceC∞# (Y;R3) of inﬁnitely smooth andY-periodic

functions.

For any elementθ ofL2₍_Ωε

e;Rn) orL2(Ωmε;Rn) we denote the extension by 0 to the

re-maining part ofΩbyθ. By taking (v,φ,ψ) =sε_{– (0,}_φ

0,ψ0) in the variational formulation of (Pε_{) we deduce that (}_uε_,_∇_φε_,∇_ψε_{) are bounded in}_H1

ΓD(Ω;R

3₎_×_L2₍_Ω_;_R3₎_×_L2₍_Ω_;_R3_). The Poincaré or Poincaré–Wirtinger inequalities and the sharp estimate of [5]

∃C(Ω) > 0 s.t.

Ωεe

|v|2dx≤C(Ω)

Ωeε

|∇v|2dx ∀v∈H_Γ1ε e,ext

(4)

(5)

(6)

Here Lu:=

Ω

f·udx+

ΓN

g·udx ∀u∈HΓ1D

Ω;R3

and

V1,1:=K1×K2×K3×K4×K5×K6, V2,1:=K1×K2×K3×K4×K6,

V1,2:=K1×K2×K7×K8, V2,2:=V1,2,

V1,3:=K1×K2×K3×K4×K8, V2,3:=V1,3,

V1,4:=K1×K2×K7×K3×K6, V2,4:=K1×K2×K7×K6,

where K1=HΓ1D

Ω;R3, K2=L2

Ω;H#1

Y;R3, K3=K5=H01(Ω),

K4=L2

Ω;H#1(Ye)/R

, K6=L2

Ω;H#1(Ym)/R

, K7=L2

Ω;H#1,m(Y)/R

,

K8=L2

Ω;H#1,e(Y)/R

,

with

H#1

G;Rnthe completion with respect to the norm of H1G;Rnof the space made of the restriction to G of the elements of C#∞

Y;Rn,

H#1,e(Y) =

ψ∈H#1(Y)s.t.ψ= 0on Ye

,

H#1,m(Y) =

φ∈H#1(Y)s.t.φ= 0on Ym

.

4 Physical interpretation

By eliminating the microscopic variablesu1_,_φ1_,_ψ1_{we can characterize the nature of the} eﬀective behavior.

Whenp= (1, 1), the body has a piezoelectromagnetic behavior with an electromagnetic coupling involving the following eﬀective piezoelectromagnetic tensor:

Mkl

eﬀ:=

Y

χe(y)Me(y)

E_sk,Ek_e+euk,∇φk·El_s,El_e+eul,∇φldy

+

Y

χm(y)Mm(y)

E_sk,E_mk+euk,∇ψk·El_s,El_m+eul,∇ψldy,

where Ek _{= (}_Ek

s,Eek,Emk), k = 1, 2, . . . , 12, is any element of a basis S3 ×R3 ×R3 and (uk,φk,ψk) are a solution to

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Find (uk_,_φk_,_ψk_{) in}_V_{:= (}_H1

#(Y;R3)/R3)×(H#1(Ye)/R)×(H#1(Ym)/R);

Yχe(y)Me(y)((Eks,Eke) + (e(uk),∇φk))·(e(u),∇φ)dy

(7)

Whenp= (1, 2), (2, 2), (2, 4), the eﬀective material is purely elastic, electricity and mag-netism are only involved in the building of the eﬀective elasticity tensor by solving cell-problems.

Whenp= (1, 3), (2, 1), (2, 3), the eﬀective material is piezoelectric, magnetism is only involved in the building of the eﬀective piezoelectric tensor.

Whenp= (1, 4), the eﬀective material is piezomagnetic and electricity is only involved in the building of the eﬀective piezomagnetic tensor.

5 Discussion and conclusions

From the mathematical point of view, the models obtained by [2,3] were derived through formal homogenization approaches such as asymptotic expansions and ﬁeld-averaging. The method presented here has the advantage of providing rigorous convergence results but also to enlightening the strategic aspects that must be taken into account by an en-gineer to design the proper multi-physical device with an eﬀective electromagnetic cou-pling. Thus to have full piezoelectromagnetic behavior each phase has to be connected and the interface has to be impermeable. As the permeability/impermeability conditions are handily obtained through an additional coating of the phases of the composite, this opens the way for future investigations that will enrich our models.

Acknowledgements

Not applicable.

Funding

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CL and TW conceived of the presented research work. CL and SO proved the main result, interpreted the result and wrote the paper. All authors read and approved the ﬁnal manuscript.

Authors’ information

Christian Licht email:[email protected]; Thibaut Weller email:[email protected].

Author details

1_{Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand.}2_{LMGC-UMR 5508, Université de} Montpellier-CC048, Montpellier, France.3_{Centre of Excellence in Mathematics, CHE, Bangkok, Thailand.}

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 29 January 2019 Accepted: 2 May 2019

References

1. Van Suchtelen, J.: Product properties: a new application of composite materials. Philips Res. Rep.27, 28–37 (1972) 2. Nguyen, T.T.: Etude des comportements statique et dynamique des composites à phases piézo-électrique et

piézo-magnétique. Thesis, Université Paris XIII, Sorbonne Paris Cité, Institut Galillée, Laboratoire des Sciences des Procédés et des Matériaux (2016)

3. Nguyen, T.T., Abdelmoula, R., Li, J., Roussigne, Y., Stashkevich, A.: Wave propagating in multilayers composed of piezoelectric and piezomagnetic layers. Composites, Part B, Eng.93, 289–301 (2016)

4. Allaire, G.: Homogenization and two-scale convergence. J. Math. Anal.23, 1482–1518 (1992)