Table Of ContentMulti-field asymptotic homogenization of thermo-piezoelectric
materials with periodic microstructure
7
Francesca Fantoni1, , Andrea Bacigalupo1, , Marco Paggi1
1 ∗ ∗
0 1 IMTSchoolforAdvancedStudiesLucca,PiazzaS.Francesco19,55100Lucca,Italy
2
January 12, 2017
n
a
J
2
Abstract
1
Thisstudyproposesamulti-fieldasymptotichomogenizationfortheanalysisofthermo-piezoelectric
] materials with periodic microstructures. The effect of the microstructural heterogeneity is taken into
i
c accountbymeansofperiodicperturbationfunctions,whichderivefromthesolutionofnonhomogeneous
s recursivecellproblemsdefinedovertheunitperiodiccell. Astrongcouplingispresentbetweenthemicro
- displacementfieldandthemicroelectricpotentialfield, sincethemechanicalandtheelectricproblems
l
r arefullycoupledintheasymptoticallyexpandedmicroscalefieldequations. Themicrodisplacement,the
t electricpotential,andtherelativetemperaturefieldshavebeenrelatedtothemacroscopicquantitiesand
m
totheirgradientsinthederiveddown-scalingrelations. Averagefieldequationsofinfiniteorderhavebeen
. obtained and theclosed formof theoverall constitutivetensors hasbeen determined forthe equivalent
t
a first-order homogenized continuum. A formal solution of such equations has been derived by means of
m an asymptotic expansion of the macro fields. The accuracy of the proposed formulation is assessed in
relation to illustrative examples of a bi-material periodic microstructure subjected to harmonic body
-
d forces,freechargedensities,andheatsources,whoseperiodicityismuchgreaterthanthecharacteristic
n microstructuralsize. Thegoodagreementobtainedbetweenthesolutionofthehomogenizedmodeland
o the finite element solution of the original heterogeneous material problem confirms the validity of the
c proposedformulation.
[
1 1 Introduction
v
1
The use of composite materials has become increasingly attractive to fulfill industry needs due to their
6
mechanical and physico-chemical properties. Namely composites can be tailored to meet specific design
3
requirements. Theyincreasestrength,durability,corrosionresistance,damagetolerance,andreduceweight.
3
They find applications in numerous engineering sectors such as for example aerospace, energy, automotive,
0
. marine, electrical, chemical, and biomedical.
1
In such a context, electromechanical devices (transducers, resonators, acoustic wave sensors among the
0
others)arenoteworthy. Theyinvolvetheuseofpiezoelectricmaterials,exhibitingalinearcouplingbetween
7
the electric and the mechanical fields. Due to their crystalline structure, which becomes electrically polar-
1
ized if mechanically stressed (direct piezoelectric effect), they also experience mechanical deformations in
:
v the presence of an electric field (inverse piezoelectric effect) (Yang, 2004). Piezoelectric materials include
i alsopyroelectricmaterials,distinguishedbythermo-electricalinteractions. Pyroelectricitycharacterizesthe
X
dependenceofspontaneouspolarizationofcrystallinematerialstotemperature. Itisthepropertywherebya
r
a changeoftemperaturegeneratesthepresenceofachargeonthesurfaceofthepyroelectricmaterialitselfand
the direction of the pyroelectric current depends on the nature of thermal gradients (Moulson and Herbert,
2003;BatraandAggarwal,2013). Acorrectcharacterizationofpiezoelectric,pyroelectric,and,moregener-
ally,temperature-dependentpropertiesofsuchmaterialsystemsisofgreatimportancetoenableapplications
insensing,transduction,actuation,energyharvesting,thermalimaging,gasanalysis,radiometry,andinthe
biomedicalfield,allowingtoreachtheoptimaldevicedesigninanyoperatingconditions. Productionofsuch
devices is expanding in small formats (nanometer scale), compatibly with microfabrication processes for
∗Correspondingauthors: Tel:+3905834326613,
E-mailaddresses: [email protected];[email protected]
1
micro-electronic applications (MEMS-based devices), since thin films offer distinct advantages for thermal
toelectricalenergyconversionoverbulksamples(Kimata,2013). Suchmaterialsaremicrostructured,being
characterized by a periodicity or a quasi-periodicity.
Multiscaletechniquesconstituteapowerfulmethodtostudythermo-piezoelectricmaterialswhosemacro-
scopic behaviour is influenced by multi-field phenomena occurring at the microscale. In order to obtain an
accurate and synthetic description of the macroscopic thermo-piezoelectric properties without performinga
computationallyexpensivemodelingofthewholeheterogeneousmedium,homogenizationtechniquescanbe
efficientlyemployed. Forcompositeswithaperiodicmicrostructure,inparticular,homogenizationrepresents
a suitable tool for modeling the effects of the microscopic phases on the overall properties. Such techniques
developed for determining the overall static and dynamic properties of periodic composite materials can be
generallydistinguishedinasympthoticmethods(Allaire,1992;Andrianovetal.,2008;AuriaultandBonnet,
1985;Bacigalupo,2014;BakhvalovandPanasenko,1984;Bensoussanetal.,1978;GambinandKro¨ner,1989;
Sanchez-Palencia,1974;Tranetal.,2012),variational-asymptotictechniques(BacigalupoandGambarotta,
2014a,b; Bacigalupo et al., 2014; Peerlings and Fleck, 2004; Smyshlyaev and Cherednichenko, 2000), and
computational approaches based on a polynomial approximation of the macroscopic fields (Addessi et al.,
2013; Bacca et al., 2013a,b,c; Bacigalupo and De Bellis, 2015; Bacigalupo and Gambarotta, 2010; Bigoni
and Drugan, 2007; Forest and Sab, 1998; Forest and Trinh, 2011; Miehe et al., 1999; Kouznetsova et al.,
2002,2004). Inalltheseformulations,theequivalenthomogenizedmacroscopicmediumisdescribedtrough
a first-order continuum or, alternatively, by means of a nonlocal one.
An extension of the above mentioned techniques to a multi-field case has concerned thermomechanics
(Aboudi et al., 2001; Kanout´e et al., 2009; Zhang et al., 2007), piezoelectricity (Deraemaeker and Nasser,
2010), and thermo-diffusive phenomena (Bacigalupo et al., 2014, 2016a,b). In (Salvadori et al., 2014) the
effects of the microstructure on macroscopic elastic and thermodiffusive properties and on the coupling be-
tween them has been investigated in relation to lithium-ion batteries and solid oxide fuel cells.
In this framework, the present work describes an asymptotic homogenization technique for static anal-
ysis of thermo-piezoelectric materials with a periodic microstructure, making a progress from the ap-
proaches proposed in (Bacigalupo, 2014; Bacigalupo and Gambarotta, 2014b; Bakhvalov and Panasenko,
1984; Smyshlyaev and Cherednichenko, 2000).
In this study, the asymptotic expansion of the micro-displacement, electric potential, and relative tem-
perature fields allows to obtain from the expression of the micro-scale field equations a series of recursive
differential problems defined over the periodic unit cell in terms of the microscopic variables. Solvability
conditionsforsuchnonhomogeneousrecursivecellproblemsleadtothederivationofdown-scalingrelations,
which correlate the microscopic fields to the macroscopic ones and to their gradients by means of pertur-
bation functions. Those functions depend exclusively on the geometrical and physico-mechanical properties
ofthematerialunderconsiderationandaccountfortheinfluenceofmicrostructuralinhomogeneitiesonthe
displacement, electric potential, and relative temperature, and on their coupling. The exact expressions of
theoverallconstitutivetensorshavebeendeterminedfortheclassofperiodicthermo-piezoelectricmaterials
considered herein, expressed in terms of perturbation functions and constitutive tensors at the microscale.
Average field equations of infinite order have been derived following the method proposed in (Bacigalupo,
2014; Smyshlyaev and Cherednichenko, 2000), substituting the down-scaling relations into the micro-field
equations. Their formal solution has been obtained by performing an asymptotic expansion of the macro
fields in terms of the characteristic microstructural size. Finally, truncating such expansions to the zeroth
order, field equations for the equivalent Cauchy thermo-piezoelectric medium are derived.
Section 2 of the paper is devoted to describe the governing equations at the microscale. Cell problems
and their solution are accurately presented in Section 3, which shows the existence of a strong coupling
between the mechanical and the electrical fields and the related perturbation functions. Section 4 presents
thedown-scalingandtheup-scalingrelations,whichallowtoexpressthemacroscopicdisplacement,electric
potential, and temperature fields in terms of the corresponding microscopic ones. Average field equations
ofinfiniteorderaredescribedinSection5,togetherwiththeclosedformoftheoverallthermo-piezoelectric
constitutive parameters and the field equations of the equivalent first-order continuum.
The homogenization technique proposed in the paper is finally applied to the case of a bi-material pe-
riodic microstructure subjected to periodic body forces, charge densities, and heat sources. The solution
of the first-order homogenized problem is then compared with the numerical results obtained by a finite
element analysis of the heterogeneous problem, in order to assess the validity of the proposed homogeniza-
tion procedure and its accuracy. In this regard, a thermo-piezoelectric element has been formulated and
2
Figure 1: (a) Periodic microstructure of the heterogeneous medium with structural characteristic size L;
(b)Periodic cell with microstructural characteristic size ε and periodicity vectors v and v ; (c) Periodic
1 2
A
unit cell .
Q
implemented in the finite element program FEAP (Zienkiewicz and Taylor, 1977) to numerically solve the
coupled thermo-electromechanical problem. Finally, conclusions are proposed in Section 7.
2 Field equations of the periodic heterogeneous material and kine-
matic multi-scale description
Consideraheterogeneouscompositematerialcharacterizedbyaperiodicmicrostructureundertheassump-
tionofsmallstrains. Thecontinuumisdescribedasalinearthermo-piezoelectricCauchymedium(Mindlin,
1974)subjecttostressesinducedbybodyforces,freechargedensities,andtemperaturechanges. Restricting
thenotationtothetwo-dimensionalcaseforthesakeofsimplicityandwithoutlosinggenerality,theposition
vectorx=x e +x e characterizeseachmaterialpointintheorthogonalreferencesystem O,e ,e with
1 1 2 2 1 2
{ }
origin at point O, as shown in Figure 1(a).
The continuum is described by the micro displacement field u(x)=u e , the micro electrical potential
i i
field φ(x), and the micro relative temperature field θ(x) = T(x) T , where T is a reference stress-free
0 0
−
temperature. Theentireperiodicmediumcanbeobtainedbyspanningaperiodiccell =[0,ε] [0,δε]by
A ×
thetwoorthogonalperiodicityvectorsdefinedasv =d e =εe ,andv =d e =δεe ,beingεthechar-
1 1 1 1 2 2 2 2
acteristicsizeofthecell (Figure1(b)). Becauseofthe -periodicityofthematerial,themicro-elasticity
A A
tensorC(m,ε)(x)=Ci(jmk,lε)ei⊗ej⊗ek⊗el,togetherwiththemicro-dielectricpermittivitytensor(atconstant
strain)β(m,ε)(x)=β(m,ε)e e ,andthemicro-heatconductiontensorK(m,ε)(x)=K(m,ε)e e obeythe
ij i⊗ j ij i⊗ j
following properties:
C(m,ε)(x+vi)=C(m,ε)(x), i=1,2, x , (1a)
∀ ∈A
β(m,ε)(x+v )=β(m,ε)(x), i=1,2, x , (1b)
i
∀ ∈A
K(m,ε)(x+v )=K(m,ε)(x), i=1,2, x , (1c)
i
∀ ∈A
where the superscript m refers to the microscale. Analogous relations hold for the micro-coupling tensors,
namely the piezoelectric stress/charge tensor e(m,ε)(x) = e(m,ε)e e e , the thermal dilatation tensor
ijk i⊗ j ⊗ k
3
α(m,ε)(x)=α(m,ε)e e , and the pyroelectric vector γ(m,ε)(x)=γ(m,ε)e :
ij i⊗ j i i
e(m,ε)(x+v )=e(m,ε)(x), i=1,2, x , (2a)
i
∀ ∈A
α(m,ε)(x+v )=α(m,ε)(x), i=1,2, x , (2b)
i
∀ ∈A
γ(m,ε)(x+v )=γ(m,ε)(x), i=1,2, x . (2c)
i
∀ ∈A
Rescalingthesizeoftheperiodiccell bythecharacteristiclengthε,itispossibletoreproducetheperiodic
A
microstructure by a non dimensional unit cell = [0,1] [0,δ] shown in Figure 1-(c). In this regard, two
Q ×
variables,themacroscopic(slow)onex ,andthemicroscopic(fast)oneξ=x/ε allowtheseparation
∈A ∈Q
ofthescalesdistinguishingfromthemacroandthemicroscales,seeforexample(Allaire,1992;Bacigalupo,
2014; Bakhvalov and Panasenko, 1984; Gambin and Kro¨ner, 1989; Peerlings and Fleck, 2004; Smyshlyaev
and Cherednichenko, 2000; Tran et al., 2012). Therefore, the constitutive tensors (1a)-(2c) are made only
dependent on the microscopic variable ξ. They result -periodic and defined over as
Q Q
C(m,ε)(x)=Cm(ξ=x/ε), β(m,ε)(x)=βm(ξ=x/ε), K(m,ε)(x)=Km(ξ=x/ε),
e(m,ε)(x)=em(ξ=x/ε), α(m,ε)(x)=αm(ξ=x/ε), γ(m,ε)(x)=γm(ξ=x/ε). (3)
Constitutive relations determining the micro-stress σ(x), the micro-electric displacement D(x), and the
micro-heat flux q(x) are given by the following formulae (Mindlin, 1974):
x x x
σ(x)=Cm ε(x)+em φ(x) αm θ(x), (4a)
ε ε ∇ − ε
(cid:2)x(cid:3) (cid:2)x(cid:3) (cid:2)x(cid:3)
D(x)=e˜m ε(x) βm φ(x)+γm θ(x), (4b)
ε − ε ∇ ε
(cid:2) (cid:3)x (cid:2) (cid:3) (cid:2) (cid:3)
q(x)= Km θ(x), (4c)
− ε ∇
(cid:2) (cid:3)
whereε=sym urepresentsthemicro-straintensorande˜m=e˜m e e e ,withe˜m =em . Underthe
∇ ijk i⊗ j⊗ k ijk jki
assumptionofaquasi-staticprocess,thetimederivativeofu(x)andθ(x)hasbeenneglectedintheheatflux
equation (4c) (Mindlin, 1974; Nowacki, 1986). The stress field σ(x), the electric displacement field D(x),
and the heat flux q(x) satisfy, respectively, the following local balance equations
σ(x)+b(x)=0, D(x) ρ (x)=0, q(x)+r(x)=0, (5)
e
∇· ∇· − ∇·
where body forces b(x), free charge densities ρ (x), and heat sources r(x) are made dependent only on
e
the slow variable x. It is assumed that volume forces are -periodic with = [0,L] [0,δL], and have
L L ×
vanishing mean values on . The structural length L has to be much greater than the microstructural
L
length ε (L >> ε) for the validity of the scales separability condition, so that can be considered as a
L
true representative portion of the whole body. The following partial differential equations result from the
substitution of constitutive relations (4a)-(4c) into the local balance equations (5)
x x x
Cm u(x) + em φ(x) αm θ(x) +b(x)=0, (6a)
∇· ε ∇ ∇· ε ∇ −∇· ε
(cid:2) (cid:2)x(cid:3) (cid:3) (cid:2) (cid:2)x(cid:3) (cid:3) (cid:2) (cid:2)x(cid:3) (cid:3)
e˜m u(x) βm φ(x) + γm θ(x) ρ (x)=0, (6b)
∇· ε ∇ −∇· ε ∇ ∇· ε − e
(cid:2) (cid:2) x(cid:3) (cid:3) (cid:2) (cid:2) (cid:3) (cid:3) (cid:2) (cid:2) (cid:3) (cid:3)
Km θ(x) +r(x)=0. (6c)
∇· ε ∇
(cid:2) (cid:2) (cid:3) (cid:3)
Denoting with [[f]] = fi(Σ) fj(Σ) the jump of the values of function f at the interface Σ between
−
two different phases i and j in the periodic cell , the following continuity conditions hold for a perfectly
A
bounded interface
x x x
[[u(x)]] =0, Cm u(x)+em φ(x) αm θ(x) n =0, (7a)
|x∈Σ ε ∇ ε ∇ − ε · x Σ
(cid:4)(cid:4)(cid:2) (cid:2)x (cid:3) (cid:2)x(cid:3) (cid:2)x(cid:3) (cid:3) (cid:5)(cid:5)(cid:6) ∈
[[φ(x)]] =0, e˜m u(x) βm φ(x)+γm θ(x) n (cid:6) =0, (7b)
|x∈Σ ε ∇ − ε ∇ ε · (cid:6)x Σ
(cid:4)(cid:4)(cid:2) (cid:2)x (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:3) (cid:5)(cid:5)(cid:6) ∈
[[θ(x)]] =0, Km θ(x) n =0, (cid:6) (7c)
|x∈Σ ε ∇ · x Σ (cid:6)
(cid:4)(cid:4) (cid:2) (cid:3) (cid:5)(cid:5)(cid:6) ∈
(cid:6)
(cid:6)
4
where n denotes the outward normal to the interface Σ. Because of the -periodicity of microscopic con-
Q
stitutive tensors in equations (6a)-(6c) and interface conditions (7a)-(7c), and taking into account the -
L
periodicityofvolumeforces,themicro-fieldsdependonboththeslowvariablexandthefastoneξ andcan
be expressed in the following form
x x x
u=u x;ξ= , φ=φ x;ξ= , θ=θ x;ξ= .
ε ε ε
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Deriving the solution of the set of PDEs (6a)-(6c) could be particularly complex both analytically and
numericallybecauseofthe -periodicityoftheircoefficients. Theuseofhomogenizationtechniquestoreplace
Q
thecontinuumwithitsmicrostructurebyanequivalenthomogeneousonecanbeveryeffective. Inthisregard,
the formulation of an equivalent first-order thermo-piezoelectric continuum will be derived in what follows
and the exact expressions of the overall constitutive tensors will be determined. This allows to replace the
solutionofequations(6a)-(6c)oftheheterogeneousmediumwiththesolutionofanequivalenthomogeneous
material whose solutions are close to those of (6a)-(6c), but whose coefficients are not affected by rapid
oscillations due to the underlying microstructure. By means of the asymptotic homogenization technique,
the computational cost of solving the microscopic field equations (6a)-(6c) can therefore be overcome and
theglobalbehaviourofthecompositestructurecanbeaccuratelydescribed. Intheequivalenthomogenized
continuum,themacroscopicfieldsineachmaterialpointxareindicatedasU(x)=U e forthedisplacement,
i i
Φ(x) for the electric potential, and Θ(x) for the relative temperature.
3 Asymptotic expansions of microscopic field equations
Thethreemicro-fieldsgoverningtheproblemcanberepresentedtroughanasymptoticexpansioninpowers
ofthemicrostructurallengthscaleεthatmantainsthemicroandthemacroscalesseparatedbykeepingthe
slow variable x distinguished from the fast one ξ = x/ε. Following the approach presented in (Bakhvalov
andPanasenko,1984;Bensoussanetal.,1978;Sanchez-Palencia,1974),theasymptoticexpansionstakethe
general form
+
u x,x = ∞εlu(l) x,x =u(0) x,x +εu(1) x,x +ε2u(2) x,x +O(ε3), (8a)
h ε h ε h ε h ε h ε
(cid:2) (cid:3) (cid:7)l=0 (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
+
x ∞ x x x x
φ x, = εlφ(l) x, =φ(0) x, +εφ(1) x, +ε2φ(2) x, +O(ε3), (8b)
ε ε ε ε ε
(cid:2) (cid:3) (cid:7)l=0 (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
+
x ∞ x x x x
θ x, = εlθ(l) x, =θ(0) x, +εθ(1) x, +ε2θ(2) x, +O(ε3). (8c)
ε ε ε ε ε
(cid:2) (cid:3) (cid:7)l=0 (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Exploiting the property D f(x;ξ = x) = ∂f + 1 ∂f = ∂f + 1f , expansions (8a)-(8c)
Dxj ε ∂xj ε ∂ξj ξ=x ∂xj ε ,j ξ=x
canbesubstitutedintothemicroscopicfield(cid:2)equations(6a(cid:3))(cid:6)-(6cε). In(cid:2)particular,e(cid:3)q(cid:6)uatεion(6a)takestheform
(cid:6) (cid:6)
(cid:6) (cid:6)
ε−2 Cimjklu(k0,l) + emijkφ(,k0) +
,j ,j
(cid:8) (cid:9)(cid:2) (cid:3) (cid:2) (cid:3) (cid:10)
∂u(0) ∂ ∂φ(0) ∂
+ ε−1⎧⎨(cid:14)Cimjkl(cid:15) ∂xkl +u(k1,l)(cid:16)(cid:17),j+ ∂xj (cid:2)Cimjklu(k0,l)(cid:3)+(cid:9)emijk(cid:18) ∂xk +φ(,k1)(cid:19)(cid:10),j+ ∂xj (cid:2)emijkφ(,k0)(cid:3)+
αm⎩θ(0) +
− ij ,j
(cid:2) (cid:3) (cid:20)
∂u(1) ∂ ∂u(0) ∂φ(1)
+ Cm k +u(2) + Cm k +u(1) + em +φ(2) +
⎧⎨(cid:14) ijkl(cid:15) ∂xl k,l(cid:16)(cid:17),j ∂xj (cid:14) ijkl(cid:15) ∂xl k,l(cid:16)(cid:17) (cid:9) ijk(cid:18) ∂xk ,k (cid:19)(cid:10),j
∂ ∂φ(0) ∂
+ ⎩ em +φ(1) αmθ(1) αmθ(0) +
∂xj (cid:9) ijk(cid:18) ∂xk ,k (cid:19)(cid:10)−(cid:2) ij (cid:3),j− ∂xj (cid:2) ij (cid:3)(cid:20)
5
∂u(2) ∂ ∂u(1) ∂φ(2)
+ ε Cm k +u(3) + Cm k +u(2) + em +φ(3) +
⎧⎨(cid:14) ijkl(cid:15) ∂xl k,l(cid:16)(cid:17),j ∂xj (cid:14) ijkl(cid:15) ∂xl k,l(cid:16)(cid:17) (cid:9) ijk(cid:18) ∂xk ,k (cid:19)(cid:10),j
∂ ∂φ(1) ∂
+ ⎩ em +φ(2) αmθ(2) αmθ(1) +O(ε2) +b (x)=0. (9)
∂xj (cid:9) ijk(cid:18) ∂xk ,k (cid:19)(cid:10)−(cid:2) ij (cid:3),j− ∂xj (cid:2) ij (cid:3)(cid:20) (cid:20)(cid:6)(cid:6)ξ=xε i
Similarly, equation (6b) can be written as (cid:6)(cid:6)
ε−2 emkliu(k0,l) ,i− βiml φ(,l0) ,i +
(cid:8) (cid:9)(cid:2) (cid:3) (cid:2) (cid:3) (cid:10)
∂u(0) ∂ ∂φ(0) ∂
+ ε−1⎧⎨(cid:14)emkli(cid:15) ∂xkl +u(k1,l)(cid:16)(cid:17),i+ ∂xi (cid:2)emkliu(k0,l)(cid:3)−(cid:9)βiml (cid:18) ∂xl +φ(,l1)(cid:19)(cid:10),i− ∂xi (cid:2)βiml φ(,l0)(cid:3)+(cid:2)γimθ(0)(cid:3),i⎫⎬
⎩ ∂u(1) ∂ ∂u(0) ∂φ(1) ⎭
+ em k +u(2) + em k +u(1) βm +φ(2) +
⎧⎨(cid:14) kli(cid:15) ∂xl k,l(cid:16)(cid:17),i ∂xi (cid:14) kli(cid:15) ∂xl k,l(cid:16)(cid:17)−(cid:9) il (cid:18) ∂xl ,l (cid:19)(cid:10),i
∂ ∂φ(0) ∂
⎩ βm +φ(1) + γmθ(1) + γmθ(0) +
− ∂xi (cid:9) il (cid:18) ∂xl ,l (cid:19)(cid:10) (cid:2) i (cid:3),i ∂xi (cid:2) i (cid:3)(cid:20)
∂u(2) ∂ ∂u(1) ∂φ(2)
+ ε em k +u(3) + em k +u(2) βm +φ(3) +
⎧⎨(cid:14) kli(cid:15) ∂xl k,l(cid:16)(cid:17),i ∂xi (cid:14) kli(cid:15) ∂xl k,l(cid:16)(cid:17)−(cid:9) il (cid:18) ∂xl ,l (cid:19)(cid:10),i
∂ ∂φ(1) ∂
⎩ βm +φ(2) + γmθ(2) + γmθ(1) +O(ε2) ρ (x)=0, (10)
− ∂xi (cid:9) il (cid:18) ∂xl ,l (cid:19)(cid:10) (cid:2) i (cid:3),i ∂xi (cid:2) i (cid:3)(cid:20) (cid:20)(cid:6)(cid:6)ξ=xε − e
and finally, eq. (6c) results (cid:6)(cid:6)
∂θ(0) ∂
(cid:24)ε−2(cid:2)Kimj θ,(j0)(cid:3),i+ε−1(cid:24)(cid:9)Kimj (cid:18) ∂xj +θ,(j1)(cid:19)(cid:10),i+ ∂xi (cid:2)Kimj θ,(j0)(cid:3)(cid:25)+
∂θ(1) ∂ ∂θ(0)
+ Km +θ(2) + Km +θ(1) +
ij ∂x ,j ∂x ij ∂x ,j
(cid:9) (cid:18) j (cid:19)(cid:10),i i (cid:9) (cid:18) j (cid:19)(cid:10)
∂θ(2) ∂ ∂θ(1)
+ ε Km +θ(3) + Km +θ(2) +
(cid:24)(cid:9) ij (cid:18) ∂xj ,j (cid:19)(cid:10),i ∂xi (cid:9) ij (cid:18) ∂xj ,j (cid:19)(cid:10)(cid:25)
+ O(ε2) +r(x)=0. (11)
ξ=x
ε
Interfaceconditions(7a)-(7(cid:26)c)(cid:6)(cid:6)canbeexpressedintermsofthefastvariableξ,sincethemicrofieldsu(x;ξ),
φ(x;ξ), and θ(x;ξ) are assumed to be -periodic regular functions of the variable x (Bakhvalov and
Q
Panasenko, 1984). Denoting with Σ the interface between two different phases in the unit cell , and
1
Q
taking into account asymptotic expansions (8a)-(8c), interface conditions (7a) are rephrased as
u(0) +ε u(1) +ε2 u(2) +...=0,
h h h
(cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1
1 (cid:6) (cid:6) (cid:6) ∂u(0) ∂φ(0)
Cm(cid:6) u(0)+em φ(0) (cid:6)n + C(cid:6)m k +u(1) +em +φ(1)
ε (cid:4)(cid:4)(cid:2) ijkl k,l ijk ,k (cid:3) j(cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:14)(cid:14)(cid:24) ijkl(cid:15) ∂xl k,l(cid:16) ijk(cid:18) ∂xk ,k (cid:19)
(cid:6)(cid:6) ∂u(1) ∂φ(1)
αmθ(0) n +ε Cm k +u(2) +em +φ(2) αmθ(1) n +
− ij (cid:27) j(cid:5)(cid:5)(cid:6)(cid:6)ξ∈Σ1 (cid:14)(cid:14)(cid:24) ijkl(cid:15) ∂xl k,l(cid:16) ijk(cid:18) ∂xk ,k (cid:19)− ij (cid:25) j(cid:17)(cid:17)(cid:6)(cid:6)(cid:6)ξ∈Σ1
∂u(cid:6)(2) ∂φ(2) (cid:6)
ε2 Cm k +u(3) +em +φ(3) αmθ(2) n +...=0. (cid:6) (12)
(cid:14)(cid:14)(cid:24) ijkl(cid:15) ∂xl k,l(cid:16) ijk(cid:18) ∂xk ,k (cid:19)− ij (cid:25) j(cid:17)(cid:17)(cid:6)(cid:6)(cid:6)ξ∈Σ1
Analogously, interface conditions (7b) read (cid:6)
(cid:6)
φ(0) +ε φ(1) +ε2 φ(2) +...=0,
(cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1
(cid:6) (cid:6) (cid:6)
(cid:6) (cid:6) (cid:6)
6
1 ∂u(0) ∂φ(0)
em u(0) βmφ(0) n + em k +u(1) βm +φ(1)
ε (cid:4)(cid:4)(cid:2) kli k,l − il ,l (cid:3) i(cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:14)(cid:14)(cid:24) kli(cid:15) ∂xl k,l(cid:16)− il (cid:18) ∂xl ,l (cid:19)
(cid:6)(cid:6) ∂u(1) ∂φ(1)
+γmθ(0) n +ε em k +u(2) βm +φ(2) +γmθ(1) n +
i (cid:27) i(cid:5)(cid:5)(cid:6)(cid:6)ξ∈Σ1 (cid:14)(cid:14)(cid:24) kli(cid:15) ∂xl k,l(cid:16)− il (cid:18) ∂xl ,l (cid:19) i (cid:25) i(cid:17)(cid:17)(cid:6)(cid:6)(cid:6)ξ∈Σ1
∂u((cid:6)2) ∂φ(2) (cid:6)
ε2 em k +u(3) βm +φ(3) +γmθ(2) n +...=0, (cid:6) (13)
(cid:14)(cid:14)(cid:24) kli(cid:15) ∂xl k,l(cid:16)− il (cid:18) ∂xl ,l (cid:19) i (cid:25) i(cid:17)(cid:17)(cid:6)(cid:6)(cid:6)ξ∈Σ1
(cid:6)
and the interface conditions (7c) become (cid:6)
θ(0) +ε θ(1) +ε2 θ(2) +...=0,
(cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1
1 (cid:6) (cid:6) ∂θ(0) (cid:6)
Km(cid:6) θ(0) n +(cid:6) Km +(cid:6) θ(1) n +
+εε(cid:4)(cid:4)(cid:2) Kijm,j (cid:3)∂θ(i1(cid:5))(cid:5)+(cid:6)(cid:6)(cid:6)ξ∈θΣ(21) (cid:9)n(cid:9)(cid:8) ij (cid:18) ∂+xjε2 ,jK(cid:19)m(cid:20) ∂i(cid:10)θ(cid:10)(2(cid:6)(cid:6)(cid:6)(cid:6))ξ∈+Σ1θ(3) n +...=0. (14)
ij ∂x ,j i ij ∂x ,j i
(cid:9)(cid:9)(cid:8) (cid:18) j (cid:19)(cid:20) (cid:10)(cid:10)(cid:6)(cid:6)ξ∈Σ1 (cid:9)(cid:9)(cid:8) (cid:18) j (cid:19)(cid:20) (cid:10)(cid:10)(cid:6)(cid:6)ξ∈Σ1
(cid:6) (cid:6)
From equations (9)-(11) one can notice t(cid:6)hat a strong coupling exists between th(cid:6)e micro-displacement
u(x;ξ) and the micro-electric potential φ(x;ξ) because the mechanical and the electric problems remain
coupled at any order ε in the asymptotically expanded microscale field equations. Hence, the proposed
homogenizationmodelcanberegardedasageneralizationofthemulti-fieldasymptotictechniqueproposed
in (Bacigalupo et al., 2016a) for thermo-diffusive materials with weak coupling between the micro/macro
fields and the single-field standard asymptotic technique for static problems, see for example (Bacigalupo,
2014;BakhvalovandPanasenko,1984). Differentialproblemsderivingfromequations(9)-(11)arenowmade
explicit at the different orders ε for both the thermal and the piezoelectric fields. They bring to the formu-
lation of cell problems described in Section 3.1.
Heat diffusion problem
Starting from the heat diffusion problem described by the field equation (11), at the order ε 2 one has the
−
following differential problem
Kmθ(0) =h(0)(x), (15)
ij ,j
,i
(cid:2) (cid:3)
with interface conditions
θ(0) =0, Kmθ(0) n =0. (16)
ij ,j i
(cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4)(cid:2) (cid:3) (cid:5)(cid:5)(cid:6)ξ∈Σ1
Solvability condition of (15) in the cl(cid:6)ass of -periodic functions, (cid:6)together with interface conditions (16),
(cid:6) Q (cid:6)
implies that h0(x) = 0, see (Bakhvalov and Panasenko, 1984), and the solution θ(0) corresponds to the
macroscopic temperature, namely:
θ(0)(x;ξ)=Θ(x), (17)
which depends only on the slow variable x.
Taking into account the expression (17) of the solution at the expansion order ε 2, at the order ε 1
− −
equation (11) yields
∂Θ
Kmθ(1) +Km =h(1)(x), (18)
ij ,j ,i ij,i ∂xj
(cid:2) (cid:3)
with interface conditions
∂Θ
θ(1) =0, Km +θ(1) n =0. (19)
Duetothe -periodicityof(cid:4)c(cid:4)omp(cid:5)o(cid:5)n(cid:6)(cid:6)(cid:6)ξe∈nΣts1Km,on(cid:9)e(cid:9)haisjh(cid:18)1∂(xx)j= K,jm(cid:19) i=(cid:10)(cid:10)0(cid:6)(cid:6)(cid:6)(cid:6)ξf∈oΣr1thesolvabilityofthedifferential
Q ij ij,i
problem (18), where () = 1 ()dξ. In consideration of equation (17), the solution θ(1) at the order ε 1
(cid:6) · (cid:7) δ · (cid:28) (cid:29) −
takes the form Q
(cid:30) ∂Θ(x)
θ(1)(x;ξ)=M(1)(ξ) , (20)
q1 ∂x
q1
7
where the perturbation function M(1) expresses the influence of the fast variable ξ= x.
q1 ε
Perturbationfunctionsare -periodicanditisassumedthattheyhavevanishingmeanvaluesovertheunit
Q
cell . Therefore, M(1)(ξ) satisfies the following normalization condition
Q q1
1
M(1) = M(1)(ξ)dξ=0, (21)
q1 δ q1
(cid:31) !Q
which is a general property imposed for all the perturbation functions.
Bearinginmindthesolutions(17)and(20)ofthedifferentialproblemsatordersε 2andε 1,respectively,
− −
equation (11) yields at the order ε0
∂2Θ ∂ ∂Θ ∂Θ
Km M(1) +θ(2) + Km +M(1) =h(2)(x), (22)
(cid:9) ij (cid:18) q1 ∂xq1∂xj ,j (cid:19)(cid:10),i ∂xi (cid:9) ij (cid:18)∂xj q1,j ∂xq1(cid:19)(cid:10)
with interface conditions
∂2Θ
θ(2) =0, Km M(1) +θ(2) n =0. (23)
Solvability condition o(cid:4)f(cid:4)the(cid:5)d(cid:5)i(cid:6)(cid:6)(cid:6)ffξ∈erΣe1ntial pr(cid:9)o(cid:9)blemij (cid:18)(22)q1en∂taxiql1s∂txhjat ,j (cid:19) i(cid:10)(cid:10)(cid:6)(cid:6)(cid:6)(cid:6)ξ∈Σ1
∂2Θ(x) ∂2Θ(x) ∂2Θ(x)
h(2)(x)= Km M(1) + Km +K M = Km +K M ,
"(cid:2) iq2 q1 (cid:3),i#∂xq1∂xq2 (cid:28) q1q2 q2j q1,j(cid:29)∂xq1∂xq2 (cid:28) q1q2 q2j q1,j(cid:29)∂xq1∂xq2
since Km M(1) =0forthedivergencetheoremandthe -periodicityofKm andM(1). Consequently,
iq2 q1 ,i Q iq2 q1
the so"lu(cid:2)tion θ(2) of(cid:3)th#e heat diffusion differential problem at the order ε 2 takes the following form
−
∂2Θ(x)
θ(2)(x;ξ)=M(2) (ξ) , (24)
q1q2 ∂x ∂x
q1 q2
thus introducing the perturbation function M(2) .
q1q2
Piezoelectric problem
For what regards the piezoelectric problem governed by field equations (9)-(10), at the order ε 2 one has
−
Cm u(0) + em φ(0) =f(0)(x),
ijkl k,l ijk ,k i
,j ,j
(cid:2) (cid:3) (cid:2) (cid:3)
emu(0) βmφ(0) =g(0)(x), (25)
kli k,l ,i− il ,l ,i
(cid:2) (cid:3) (cid:2) (cid:3)
with interface conditions
u(0) =0, φ(0) =0,
k
(cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1
Cm (cid:6)u(0)+em φ(0) n (cid:6) =0, emu(0) βmφ(0) n =0. (26)
(cid:4)(cid:4)(cid:2) ijkl(cid:6) k,l ijk ,k (cid:3) j(cid:5)(cid:5)(cid:6)(cid:6)ξ∈Σ1 (cid:4)(cid:4)(cid:2) kli k,l − il ,l (cid:3) i(cid:5)(cid:5)(cid:6)ξ∈Σ1
(cid:6) (cid:6)
Analogouslytoheatconduction,thesolvability(cid:6) conditionfora -periodicfunctionim(cid:6)pliesthatf(0)(x)=0
Q i
and g(0)(x)= 0, and therefore the solution of problem (25) does not depend on the fast variable ξ, taking
the form
u(0)(x;ξ)=U (x), φ(0)(x;ξ)=Φ(x). (27)
k k
In force of solutions (27) of the piezoelectric differential problem at the order ε 2, from equations (9) and
−
(10), at the order ε 1 one has
−
∂U ∂Φ
Cm u(1) +Cm k + em φ(1) +em αm Θ=f(1)(x),
ijkl k,l ,j ijkl,j ∂xl ijk ,k ,j ijk,j ∂xk − ij,j i
(cid:2) (cid:3) (cid:2) (cid:3)
∂U ∂Φ
em u(1) +em k βmφ(1) βm +γmΘ=g(1)(x), (28)
kli k,l ,i kli,i ∂xl − il ,l ,i− il,i ∂xl i,i
(cid:2) (cid:3) (cid:2) (cid:3)
8
with interface conditions
u(1) =0, φ(1) =0,
k
(cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1 (cid:4)(cid:4) (cid:5)(cid:5)(cid:6)ξ∈Σ1
Cimjk(cid:6)(cid:6)l ∂∂Uxk +u(k1,l) +emi(cid:6)(cid:6)jk ∂∂xΦ +φ(,k1) −αimjΘ nj =0,
(cid:9)(cid:9)(cid:18) (cid:18) l (cid:19) (cid:18) k (cid:19) (cid:19) (cid:10)(cid:10)(cid:6)(cid:6)ξ∈Σ1
emkli ∂∂Uxk +u(k1,l) −βiml ∂∂xΦ +φ(,l1) +γimΘ ni (cid:6)(cid:6) =0. (29)
(cid:9)(cid:9)(cid:18) (cid:18) l (cid:19) (cid:18) l (cid:19) (cid:19) (cid:10)(cid:10)(cid:6)(cid:6)ξ∈Σ1
In this case, the solvability condition implies that functions f(1)(x) and g(1)(x(cid:6)(cid:6)) take the form
i
∂U (x) ∂Φ(x)
f(1)(x)= Cm k + em αm Θ(x)=0,
i ijkl,j ∂x ijk,j ∂x − ij,j
l k
(cid:28) (cid:29) ∂U (x) (cid:28) (cid:29)∂Φ(x) (cid:28) (cid:29)
g(1)(x)= em k βm + γm Θ(x)=0,
kli,i ∂x − il,i ∂x i,i
l l
and, consequently, the solutions o(cid:28)f the d(cid:29)ifferential pr(cid:28)oblem(cid:29) (28) read(cid:28) (cid:29)
∂U (x) ∂Φ(x)
u(1)(x;ξ)=N(1) (ξ) p +N˜(1)(ξ) +Nˆ(1)(ξ)Θ(x),
k kpq1 ∂xq1 kq1 ∂xq1 k
∂Φ(x) ∂U (x)
φ(1)(x;ξ)=W(1)(ξ) +W˜(1)(ξ) p +Wˆ(1)(ξ)Θ(x). (30)
q1 ∂x pq1 ∂x k
q1 q1
Finally,takingintoaccountsolutions(27)and(30)ofthepiezoelectricdifferentialproblemsatε 2 andε 1,
− −
differential problems (9)-(10) at the order ε0 are expressed as
∂2U
+ Cm u(2) + Cm N(1) +Cm +Cm N(1) + em W˜(1) +em W˜(1) p
(cid:2) ijkl k,l(cid:3),j (cid:9)(cid:2) ijkq2 kpq1(cid:3),j iq1pq2 iq2kl kpq1,l (cid:2) ijq2 pq1(cid:3),j iq2k pq1,k(cid:10)∂xq1∂xq2
∂2Φ
+ em φ(2) + Cm N˜(1) +Cm N˜(1) + em W(1) +em +em W(1)
(cid:2) ijk ,k (cid:3),j (cid:9)(cid:2) ijkq2 kq1(cid:3),j iq2kl kq1,l (cid:2) ijq2 q1 (cid:3),j iq1q2 iq2k q1,k(cid:10)∂xq1∂xq2
∂Θ
+ Cm Nˆ(1) +Cm Nˆ(1)+ em Wˆ(1) +em Wˆ(1) αmM(1) αm =f(2)(x),
(cid:9)(cid:2) ijkq1 k (cid:3),j iq1kl k,l (cid:2) ijq1 (cid:3),j iq1k ,k −(cid:2) ij q1 (cid:3),j− iq1(cid:10)∂xq1 i
∂2U
+ em u(2) + em N(1) +em N(1) +e βmW˜(1) βm W˜(1) p
(cid:2) kli k,l(cid:3),i (cid:9)(cid:2) kq2i kpq1(cid:3),i klq2 kpq1,l pq2q1−(cid:2) iq2 pq1(cid:3),i− q2l pq1,l(cid:10)∂xq1∂xq2
∂2Φ
βmφ(2) + em N˜(1) +em N˜(1) βmW(1) βm βm W(1)
−(cid:2) il ,l (cid:3),i (cid:9)(cid:2) kq2i kq1(cid:3),i klq2 kq1,l−(cid:2) iq2 q1 (cid:3),i− q1q2− q2l q1,l(cid:10)∂xq1∂xq2
∂Θ
+ em Nˆ(1) +em Nˆ(1) βmWˆ(1) βm Wˆ(1)+ γmM(1) γm =g(2)(x), (31)
(cid:9)(cid:2) kq1i k (cid:3),i klq1 k,l −(cid:2) iq1 (cid:3),i− q1l ,l (cid:2) i q1 (cid:3),i− q1(cid:10)∂xq1
with interface conditions
u(2) =0, φ(2) =0,
k
(cid:4)(cid:4) C(cid:5)(cid:5)m(cid:6)(cid:6)(cid:6)ξ∈uΣ(12) + e(cid:4)m(cid:4) φ(2(cid:5))(cid:5)(cid:6)(cid:6)(cid:6)ξ+∈Σ1Cm N(1) +em W˜(1) ∂2Up + Cm N˜(1) +em W(1) ∂2Φ
(cid:9)(cid:9)(cid:8)(cid:2) ijkl k,l(cid:3) (cid:2) ijk ,k (cid:3) (cid:2) ijkl kpq1 ijl pq1(cid:3)∂xq1∂xl (cid:2) ijkl kq1 ijl q1 (cid:3)∂xq1∂xl
∂Θ
+ Cm Nˆ(1)+em Wˆ(1) α M(1) n =0,
ijkl k ijl − ij l ∂x j
(cid:2) (cid:3) l(cid:20) (cid:10)(cid:10)(cid:6)(cid:6)ξ∈Σ1
em u(2) βmφ(2) + em N(1) βmW˜(cid:6)(cid:6)(1) ∂2Up + em N˜(1) βmW(1) ∂2Φ
(cid:9)(cid:9)(cid:8)(cid:2) kli k,l(cid:3)−(cid:2) il ,l (cid:3) (cid:2) kli kpq1− il pq1(cid:3)∂xq1∂xl (cid:2) kli kq1− il q1 (cid:3)∂xq1∂xl
∂Θ
+ em Nˆ(1) βmWˆ(1)+γ M(1) n =0. (32)
kli k − il i l ∂x i
(cid:2) (cid:3) l(cid:20) (cid:10)(cid:10)(cid:6)(cid:6)ξ∈Σ1
Here, the following relations hold because of the sol(cid:6)vability condition for problem (31)
(cid:6)
∂2U (x)
f(2)(x)= Cm +Cm N(1) +em W˜(1) p +
i (cid:31) iq1pq2 iq1pq2 kpq1,l iq2k pq1,k ∂xq1∂xq2
9
∂2Φ(x)
+ Cm N˜(1) +em +em W(1) +
(cid:31) iq2kl kq1,l iq1q2 iq2k q1,k ∂xq1∂xq2
∂Θ(x)
+ Cm Nˆ(1)+em Wˆ,k(1) αm ,
iq1kl k,l iq1k − iq1 ∂x
(cid:31) q1
∂2U (x)
g(2)(x)= em N(1) +em βm W˜(1) p +
(cid:31) klq2 kpq1,l pq2q1− q2l pq1,l ∂xq1∂xq2
∂2Φ(x)
+ em N˜(1) βm βm W(1) +
(cid:31) klq2 kq1,l− q1q2− q2l q1,l ∂xq1∂xq2
∂Θ(x)
+ em Nˆ(1) βm Wˆ,l(1)+γm ,
klq1 k,l − q1l q1 ∂x
(cid:31) q1
the solutions u(2) and φ(2) take the following form
k
∂2U ∂2Φ ∂Θ
u(2)(x;ξ)=N(2) (ξ) p +N˜(2) (ξ) +Nˆ(2)(ξ) ,
k kpq1q2 ∂xq1∂xq2 kq1q2 ∂xq1∂xq2 kq1 ∂xq1
∂2Φ ∂U ∂Θ
φ(2)(x;ξ)=W(2) (ξ) +W˜(2) (ξ) p +Wˆ(2)(ξ) . (33)
q1q2 ∂x ∂x pq1q2 ∂x ∂x q1 ∂x
q1 q2 q1 q2 q1
Thegeneralexpressionofhigherordersolutionsu(m)(x;ξ),φ(m)(x;ξ)andθ(m)(x;ξ),withm Zandm 1,
k ∈ ≥
is reported in Appendix A.
3.1 Cell problems and perturbation functions
Solutions of recursive differential problems (9)-(11) allow to write non-homogeneous cell problems at the
different orders of ε in terms of the perturbation functions. Perturbation functions exclusively depend on
geometrical and physico-mechanical properties of the material reflecting the effects of the material inho-
mogeneities on displacements, electric potential and temperature. In the following, the form of such cell
problems is discussed at the different orders of ε, for heat diffusion and piezoelectric problems.
Heat diffusion problem
From equation (18) and in consideration of the solution (20), the cell problem at the order ε 1 takes the
−
form
KmM(1) +Km =0, (34)
ij q1,j ,i iq1,i
(cid:2) (cid:3)
with interface conditions expressed in terms of perturbation function M(1) as
q1
M(1) =0,
(cid:4)(cid:4) q1 (cid:5)(cid:5)(cid:6)ξ∈Σ1
Km M(cid:6)(1) +δ n =0. (35)
(cid:4)(cid:4) ij (cid:2) (cid:6)q1,j jq1(cid:3) i(cid:5)(cid:5)(cid:6)ξ∈Σ1
(cid:6)
OncetheperturbationfunctionM(1) isdetermined,fromequatio(cid:6)n(22)andtakingintoaccountthesolution
q1
(24), one obtains the cell problem at the order ε0, where symmetrization with respect to indices q and q
1 2
is introduced. This procedure leads to
1
KmM(2) + Km M(1) +Km +Km M(1) + Km M(1) +Km +Km M(1) =
ij q1q2,j ,i 2 iq2 q1 ,i q1q2 q2j q1,j iq2 q2 ,i q2q1 q1j q2,j
(cid:2) (cid:3) (cid:9)(cid:2) (cid:3) (cid:2) (cid:3) (cid:10)
1
Km +Km M(1) +Km +Km M(1) , (36)
2(cid:6) q1q2 q2j q1,j q2q1 q1j q2,j (cid:7)
with interface conditions in terms of perturbation functions taking the following form
M(2) =0,
(cid:4)(cid:4) q1q2(cid:5)(cid:5)(cid:6)ξ∈Σ1
(cid:6)
(cid:6)
10
