On Hashin-Shtrikman-type bounds for nonlinear conductors

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On Hashin-Shtrikman-type bounds for nonlinear

conductors

Michaël Peigney

To cite this version:

Michaël Peigney. On Hashin-Shtrikman-type bounds for nonlinear conductors. Comptes Rendus

Mécanique, Elsevier Masson, 2017, 345, pp.353 - 361. �10.1016/j.crme.2017.02.006�. �hal-01518063�

Contents lists available atScienceDirect

Comptes

Rendus

Mecanique

www.sciencedirect.com

On

Hashin–Shtrikman-type

bounds

for

nonlinear

conductors

Michaël Peigney

UniversitéParis-Est,LaboratoireNavier(UMR8205),CNRS,ÉcoledespontsParisTech,IFSTTAR,77455Marne-la-Vallée,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received16January2017 Accepted28February2017 Availableonline18March2017 Keywords:

Compositematerials Nonlinearhomogenization Bounds

Translationmethod

Forlinearcompositeconductors,itisknownthatthecelebratedHashin–Shtrikmanbounds canberecoveredbythetranslationmethod.Weinvestigatewhetherthesameconclusion extendsto nonlinearcompositesin twodimensions. Tothat purpose,weconsider two-phasecompositeswithperfectlyconductinginclusions.Inthatcase,explicitexpressionsof the variousboundsconsidered canbeobtained. Thebounds providedbythe translation methodarecomparedwiththenonlinearHashin–Shtrikman-typeboundsdeliveredbythe Talbot–Willis(1985)[2]andthePonteCastañeda(1991)[3]procedures.

©2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Inaseminalwork[1],HashinandShtrikmanhaveobtainedoptimalboundsontheeffectiveconductivityoflinear com-posite conductors withstatisticallyisotropicmicrostructures. Those boundsare explicitfunctions ofthevolume fractions andconductivities ofeach constitutive phase. Several methods havebeen proposed to extend the resultsof Hashin and Shtrikmantononlinearcomposites.Afirstmethod,proposedbyTalbotandWillis[2],makesusesofa

homogeneous linear

comparisonmediumandgeneralizesthevariationalapproachintroducedbyHashinandShtrikman.Asecondmethod,due toPonteCastañeda[3],employsa

heterogeneous linear

comparisonmedium,i.e.alinearcomparisoncomposite.Usingthat last method,anybound onthe effectiveconductivityofthe linearcomparisoncomposite can beused togenerate a cor-respondingbound forthenonlinearcomposite.Inparticular, whenthelinearHashin–Shtrikmanbound isused,nonlinear Hashin–Shtrikman-typeboundsareobtained. Athirdmethod,knownhasthetranslationmethod[4],hasbeenintroduced independentlybyLurieandCherkaev[5],MuratandTartar[6].Originallyintroducedinthelinearcontext,thatmethodhas proved to be very fruitfulin a lotof nonlinearhomogenization problems[7–12]. Fornonlinear isotropic conductors,the translationmethod hasbeen usedto obtainexplicit boundsforcomposites governed by threshold-typeenergy functions

[13,14].

ThethreemethodsmentionedabovecangeneratenonlinearboundsoftheHashin–Shtrikmantype,i.e.boundsthathold forthe wholeclass ofisotropic compositeswithprescribed volume fractionsandconduction propertiesofthe individual phases. As those methods are not mathematically equivalent, it is importantto understand the relations betweenthem. The relation betweenthe Talbot–Willisand the Ponte Castañeda methods has been investigated in [15]: for two-phase compositeswithperfectlyconductinginclusions,thetwomethodshavebeenprovedtogivethesameresultsiftheenergy functionofthematrixsatisfiesacertain strongconvexity condition.Ifthat conditionisviolated,theTalbot–Willismethod leads tostrongerboundsthanthePonte Castañedamethod.The relationswiththetranslationmethodhavebeenstudied in[13]:Boundsobtainedfromthetranslationmethodhavebeenprovedtobealwaysatleastasgoodasthoseprovidedby

E-mailaddress:[email protected].

http://dx.doi.org/10.1016/j.crme.2017.02.006

1631-0721/©2017Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

354 M. Peigney / C. R. Mecanique 345 (2017) 353–361

thePonteCastañedamethod.Fortwo-dimensionalcompositesgovernedbythreshold-typeenergyfunctions,thetranslation methodactuallygivesthesameboundsasthePonteCastañedamethod.Numericalresultssuggestthatthesameconclusion extendstopower-lawcomposites,althougharigorousproofisstilllacking.

Theobjectiveofthispaperistofillsomeofthegapsintherelationsbetweenthetranslationmethodandthemethods ofPonteCastañeda,TalbotandWillisbyinvestigatingconditionsunderwhichthetranslationmethodmaybringagenuine improvement.Tothatpurpose,weconsidertwo-phasecompositeswithperfectlyconductinginclusions,intwodimensions. The formulation ofthe translationmethod in that context is presented in Sect. 2. If the energy function of the matrix satisfiesthestrongconvexityassumptionintroducedin[15],weproveinSect.3thatthetranslationmethodgivesthesame boundsastheTalbot–WillisandPonteCastañedamethods.Thisconfirmsthenumericalobservationsmadein[13]forthe specialcaseofpower-lawcomposites.Forthetranslationmethodtobringagenuineimprovement,itisthereforenecessary that the energyinthe matrixisnot strongly convex.Building onthat observation, inSect.4 we providean examplefor whichboundsobtainedfromthetranslationmethodareindeedstrongerthantheTalbot–WillisandPonteCastañedabounds. 2. Bounding the effective energy via the translation method

Consider atwo-dimensionalinhomogeneouselectricconductoroccupyingadomain



ofunitvolume.Theelectricfield

e andthecurrentdensity j arerelatedbythelocalconstitutivelaw j

=

∂

w

∂

e

(

e

,

x

)

(1)

where theconvexenergy-densityfunction w depends onthelocation x. Denoting by e (resp.

¯

¯j

) thespatial averageofe

(resp. j),theeffectiveconstitutivelawreadsas[16,17]

¯j =

dweff

de

¯

(

e

¯

)

(2)

where weff istheeffectiveenergyfunctionofthecompositematerial,definedby

weff

(

e

¯

)

=

inf

e∈K(¯e)





w

(

e

,

x

)

d

ω

(3)

In(3),K

(

e

¯

)

isthesetofadmissibleelectricfields,asdefinedby

K

(

e

¯

)

= {

e

:  → R

2

|

e

= ∇

V for some V

:  → R

verifying V

(

x

)

= ¯

e

·

x on

∂

}

Following[5,6],alowerboundon

w

effcanbeobtainedbyembeddingtheoriginalprobleminaproblemofdimension4.

Inmoredetail,weintroduceextendedfieldsE

(

x

)

= (

e1

(

x

),

e2

(

x

))

obtainedbyconsidering2electricfieldse1

(

x

)

ande2

(

x

)

,

writtensidebyside.Introducingthe

extended energy

W

(

E

,

x

)

=

w

(

e1

,

x

)

+

w

(

e2

,

x

)

aswellasthe

extended effective energy

Weff

( ¯

E

)

=

weff

(

e

¯

1

)

+

weff

(

e

¯

2

)

(4)

itcanbereadilyseenfrom(3)that

Weff

( ¯

E

)

=

inf

E∈K( ¯E)





W

(

E

,

x

)

d

ω

(5)

where E

¯

= (¯

e1

,

e

¯

2

)

and

K( ¯

E

)

= {(

e1

,

e2

)

:

ei

∈

K

(

e

¯

i

)

}

.We now proceed tobound Weff frombelow. Tothat purpose,it is

convenienttorepresentextendedfieldsE

= (

e1

,

e2

)

by2

×

2 matrices,i.e.

E

=



u1 u2

v1 v2



where

u

iandviarethecomponentsoftheelectricfieldei inareferencebasisof

R

2.Foranyscalar

α

andanyT in

R

2×2,

considertheLegendretransform

(

W

−

α

det

)

∗

(

T

,

x

)

=

sup

E

E

:

T

−

W

(

E

,

x

)

+

α

det E (6)

where E

:

T

=

tr E T anddet isthedeterminantin

R

2,i.e.det E

=

u1v2

−

u2v1.ForanyE

∈

K( ¯

E

)

,itfollowsfrom(6)that

Any E

∈

K( ¯

E

)

satisfiesthewell-knownidentities



_E

(

x

)

d

ω

= ¯

E and



_det E

(

x

)

d

ω

=

detE.

¯

Hence,integrating(7)overthe domain



yields



 W

(

E

,

x

)

d

ω

≥ ¯

E

:

T

+

α

detE

¯

−





(

W

−

α

det

)

∗

(

T

,

x

)

d

ω

(8)

foranyE

∈

K( ¯

E

)

.Takingtheinfimumoverfields E in

K( ¯

E

)

gives

Weff

( ¯

E

)

≥ ¯

E

:

T

+

α

detE

¯

−





(

W

−

α

det

)

∗

(

T

,

x

)

d

ω

(9)

Theright-handsideof(9)isthusalowerboundontheextendedenergy

W

eff.Alowerboundon

w

eff canbededucedfrom (9)ifthecompositeisisotropic.Insuchcase, weffindeeddependsontheelectricfield e only

¯

throughitsnorme,

¯

i.e.there

exists afunction

φ

eff suchthat weff

(

e

¯

)

= φ

eff

(

e

¯

)

.Consequently,we have Weff

(

e

¯

1

,

e

¯

2

)

= φ

eff

(

¯

e1

)

+ φ

eff

(

e

¯

2

)

wheree

¯

i

= ¯

ei



.

Specializing(9)to E of

¯

theformeN with

¯

e

¯

≥

0 and

N

=



0 1 1 0



wethusobtain weff

(

e

¯

)

≥

1 2

⎛

⎝¯

eN

:

T

−

α

e

¯

2

−





(

W

−

α

det

)

∗

(

T

,

x

)

d

ω

⎞

⎠

(10)

Thebestbound

w

_

(

e

¯

)

deliveredbythepresentedprocedure(knownasthetranslationmethod)isobtainedby maximiz-ingtheright-handsideof(10)withrespectto

(

T

,

α

)

,i.e.

w_

(

e

¯

)

=

sup T,α 1 2

⎛

⎝¯

eN

:

T

−

α

e

¯

2

−





(

W

−

α

det

)

∗

(

T

,

x

)

d

ω

⎞

⎠

(11)

Thebound

w

isrelevantonlyif

w

(

e

,

x

)

hasfasterthanquadraticgrowthine (soastoensurethat

(

W

−

α

det

)

∗

(

T

,

x

)

<

∞

).Foranenergyfunction w

(

e

,

x

)

thatisquadraticine,thebound(11)isknowntocoincidewiththeHashin–Shtrikman bound[5,6].

Intherestofthepaper,weconsidertwo-phasecompositesconstitutedofperfectlyconductinginclusionsinanonlinear matrix.Ata pointx intheperfectlyconductingphase,theenergy w

(

e

,

x

)

vanishes ife

=

0 andisequalto

+∞

ife

=

0, sothat

(

W

−

α

det

)

∗

(

T

,

x

)

=

0.Toalleviatethenotations,fromnowonwedropthex dependenceoftheenergyfunction: fromhereon, w (resp. W ) simplydenotestheenergyfunction(resp.extended energyfunction)ofthematrix.Thebound

(11)becomes w_

(

e

¯

)

=

sup T,α w_

(

e

¯

;

T

,

α

)

(12) where w_

(

e

¯

;

T

,

α

)

=

1 2

¯

eN

:

T

−

α

e

¯

2

−

c

(

W

−

α

det

)

∗

(

T

)



(13) In(13),

c is

thevolumefractionofthematrix.

3. Two-phase composite with a strongly convex energy function

Inthissection, we assume that w is isotropic andstrongly convex, in thesense that w is convexin e2_{.Moreover, we}

assume that w is differentiableandincreasing in

e with

alarger thanquadraticgrowth.Anexample ofenergyfunctions satisfyingthoseassumptionsisprovidedbypower-lawfunctions

σ

en+1with

n

>

1 and

σ

>

0.

Asaconsequenceofthestatedassumptionson w, wecanwritethederivative

w



(

e

)

as

w

(

e

)

=

h

(

e

)

e

where h is a positive, unbounded, monotonically increasing function of the norm e. For later reference, we record the expressionoftheTalbot–Willisbound:

wTW

(

e

¯

)

=

c w



¯

e

√

2

−

c c

(14)

356 M. Peigney / C. R. Mecanique 345 (2017) 353–361

Theresult(14)canalsobeobtainedbyusingthePonteCastañedamethodinconjunctionwiththeHashin–Shtrikmanbound

[1]forthelinearcomparisoncomposite[18].

Thegoalofthissectionistoshowthatthetranslatedbound(13)alsocoincideswith(14).Thisisaccomplishedbyfinding values

(

τ

˜

,

α

˜

)

such that w_

(

¯

e

;

τ

˜

N

,

α

˜

)

=

wTW

(

e

¯

)

andsubsequentlyshowingthattheobtainedvalues

(

τ

˜

N

,

α

˜

)

maximizethe

function

(

T

,

α

)

→

w_

(

e

¯

;

T

,

α

)

. 3.1. Bounds obtained for TTT parallel to NNN

ForamatrixT oftheformT

=

τ

N ,wehave

(

W

−

α

det

)

∗

(

τ

N

)

=

sup E=u1u2 v1v2 f

(

E

;

τ

,

α

)

(15) where f

(

E

;

τ

,

α

)

=

τ

(

v1

+

u2

)

−

w

(

e1

)

−

w

(

e2

)

+

α

(

u1v2

−

u2v1

)

(16)

Setting

h

i

=

h

(

ei

)

,thestationarityconditionsin(16)readas

−

h1u1

+

α

v2

=

0

,

τ

−

h1v1

−

α

u2

=

0

−

h2v2

+

α

u1

=

0

,

τ

−

h2u2

−

α

v1

=

0 (17)

Asaresultofthenonconvexityofthefunction f

(

·;

τ

,

α

)

,thesystem(17)generallyadmitsmultiplesolutions.Limitingthe discussiontothecase

τ

,

α

≥

0,acloseinspectionshowsthattherearethreebranchesofsolutionsto(17):

– branch1correspondstosolutionsoftheform

E

=



0 v v 0



with

τ

=

h

(

v

)

v

+

α

v (18)

Since thefunction v

→

h

(

v

)

v

+

α

v is strictlyincreasing from0 to

∞

(in thedomain v

≥

0), thereisaunique v

≥

0 that satisfiestherelation

τ

=

h

(

v

)

v

+

α

v in (18).Thevaluetakenby f

(

·;

τ

,

α

)

forthematrix E in(18)isdenotedby

f1

(

τ

,

α

)

;

– branch2correspondstosolutionsoftheform

E

=



0 u2 v1 0



with u2

=

v1

,

τ

=

h1v1

+

α

u2

=

h2u2

+

α

v1 (19)

Wedenoteby f2

(

τ

,

α

)

themaximumvaluetakenby f

(

·;

τ

,

α

)

overE oftheform(19);

– branch3correspondstosolutionsoftheform

E_±

=



±

u1 ₂τ_α τ 2α

±

u1



with h

⎛

⎝



u2₁

+

τ

2 4

α

2

⎞

⎠ =

α

(20)

Such E_±existsaslongas

h

(

0

)

≤

α

and

τ

≤

τ

c

(

α

)

.Here

τ

c

(

α

)

≥

0 isthescalardefinedimplicitlyby

h



τ

c

(

α

)

2

α



=

α

(21)

Itcaneasilybeverifiedthat f

(

E₊

;

τ

,

α

)

=

f

(

E₋

;

τ

,

α

)

.Thatcommonvalue,denotedby f3

(

τ

,

α

)

,canbeexpressedin

termsof

τ

c

(

α

)

as f3

(

τ

,

α

)

=

τ

2 2

α

+

τ

2 c

(

α

)

4

α

−

2w



τ

c

(

α

)

2

α

¯

e

¯

e



Thefunction

(

W

−

α

det

)

∗

(

τ

N

)

isthemaximumofthethreebranchesdetailedabove,i.e.

(

W

−

α

det

)

∗

(

τ

N

)

=

max

1≤i≤3fi

(

τ

,

α

)

(22)

Ageneralexplicitexpressionof

(

W

−

α

det

)

∗

(

τ

N

)

–holdingforarbitraryvaluesof

(

τ

,

α

)

andanyformoftheenergy

w

–remainsoutofreach.Whatcanbeproved,however,isthat

(

W

−

α

det

)

∗

(

τ

N

)

=

f3

(

τ

,

α

)

for 0

≤

τ

≤

τ

c

(

α

)

and h

(

0

)

≤

α

(23)

Letusjustifytheproperty(23).Weconsider

α

≥

h

(

0

)

asfixedanddenoteby

v

(

τ

)

≥

0 thesolutiontotheequation

sothat f1

(

τ

,

α

)

=

2

τ

v

(

τ

)

−

2w



v

(

τ

)

e

¯

¯

e



−

α

v

(

τ

)

2 (25)

Itcaneasilybeverifiedfrom(21)and(24)that

v

(

τ

c

(

α

))

=

τ

c

(

α

)

2

α

For

v

≥

0,thefunction

v

→

h

(

v

)

v

+

α

v is strictlyincreasingfrom0 to

∞

.Hence

τ

→

v

(

τ

)

isincreasingwith

τ

,sothat

v

(

τ

)

≤

v

(

τ

c

(

α

))

=

τ

c

(

α

)

2

α

for 0

≤

τ

≤

τ

c

(

α

)

(26)

Using(24)and(25),weobtainbydifferentiation

∂

∂

τ

(

f1

−

f3

)(

τ

,

α

)

=

2v

(

τ

)

−

τ

α

=

v

(

τ

)

α

α

−

h

(

v

(

τ

))



(27) Inviewof(26)andrecallingthat

h is

increasing,wehave

h

(

v

(

τ

))

≤

h

(

τc(α)

2α

)

=

α

for

τ

≤

τ

c

(

α

)

.Hence

0

≤

∂

∂

τ

(

f1

−

f3

)(

τ

,

α

)

for 0

≤

τ

≤

τ

c

(

α

)

Thisshowsthatthefunction

τ

→ (

f1

−

f3

)(

τ

,

α

)

isincreasingon

[

0

,

τ

c

(

α

)

]

.Hence,

(

f1

−

f3

)(

τ

,

α

)

≤ (

f1

−

f3

)(

τ

c

(

α

),

α

)

for 0

≤

τ

≤

τ

c

(

α

)

For

τ

=

τ

c

(

α

)

, it can easily be verified that the matrices E in (18) and E± in (20) coincide and are equal to τc2(α N .α) Therefore,

(

f1

−

f3

)(

τ

c

(

α

),

α

)

=

0.Wethusobtainthat

(

f1

−

f3

)(

τ

,

α

)

≤

0 for 0

≤

τ

≤

τ

c

(

α

)

Asimilarreasoningcanbeusedtoshowthat

(

f2

−

f3

)(

τ

,

α

)

≤

0 for 0

≤

τ

≤

τ

c

(

α

)

Substitutingin(22)givesthedesiredresult(23).

2

Theproperty(23)allowsustoevaluatethebound

w



(

e

¯

;

τ

N

,

α

)

forany

(

τ

,

α

)

thatsatisfytheconditions0

≤

τ

≤

τ

c

(

α

)

and

h

(

0

)

≤

α

.Inparticular,considerthespecialvaluesdefinedby

˜

α

=

h



¯

e

√

2

−

c c

,

τ

˜

2

α

˜

=

¯

e c (28)

Since h is increasing, we necessarily have

h

(

0

)

≤ ˜

α

. Wealso note from (28)and(21) that

τ

c

(

α

˜

)/

2

α

˜

= ¯

e

√

2

−

c

/

c, hence

τ

c

(

α

˜

)

≥ ˜

τ

.Useof(23)yields

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

=

f3

(

τ

˜

,

α

˜

)

= (

4

−

c

)

α

˜



¯

e c



2

−

2w



¯

e

√

2

−

c c

Substitutingin(13)gives w_

(

e

¯

; ˜

τ

N

,

α

˜

)

=

1 2

2

τ

˜

¯

e

− ˜

α

e

¯

2

−

c f3

(

τ

˜

,

α

˜

)



=

c w



¯

e

√

2

−

c c

(29) whichshowsthatthetranslatedbound w_

(

e

¯

)

in(12)isatleastasgoodastheTalbot–Willisbound wTW

(

e

¯

)

in(14).Inthe

nextsection, weshowthatthevalues

(

τ

˜

N

,

α

˜

)

areinfactoptimalin(12),i.e.that thetranslatedbound w_

(

e

¯

)

isequalto wTW

(

e

¯

)

.

358 M. Peigney / C. R. Mecanique 345 (2017) 353–361

3.2. Optimized bound

Observe that the bound(12) is definedby a concave optimization problem over

(

T

,

α

)

. The function

(

W

−

α

det

)

∗

(

T

)

is indeedconvexin

(

T

,

α

)

,sinceit isdefinedin(6)asthepointwisesupremumofa familyoflinearfunctionsin

(

T

,

α

)

. Hencethevaluesof

(

T

,

α

)

reachingthesupremumin(12)arefullycharacterizedbytheoptimalityconditions

1

c

(

eN

¯

,

−¯

e

2

₎

_{∈ ∂(}

_W

₋

_α

_det

₎

∗

₍

_T

₎

₍₃₀₎

where

∂(

W

−

α

det

)

∗

(

T

)

isthe subdifferential ofthe convex function

(

T

,

α

)

→ (

W

−

α

det

)

∗

(

T

)

. We recall that vectors

(

A

,

a

)

inthesubdifferential

∂(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

arecharacterizedbytheproperty[19,20]

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

+

A

: (

T

− ˜

τ

N

)

+

a

(

α

− ˜

α

)

≤ (

W

−

α

det

)

∗

(

T

)

(31) forany

(

T

,

α

)

.

Wenowshowthat

(

τ

˜

N

,

α

˜

)

satisfiestheoptimalitycondition(30).Let E

˜

_± bethevaluetakenbythematrix E_± in(20)

forthecase

(

τ

,

α

)

= ( ˜

τ

,

α

˜

)

,i.e.

˜

E_±

=

e

¯

c



±

√

1

−

c 1 1

±

√

1

−

c



(32) Since

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

=

f3

(

τ

˜

,

α

˜

)

,weknowthat

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

= ˜

τ

E

˜

₊

:

N

+ ˜

α

detE

˜

₊

−

W

( ˜

E₊

)

ForanyT and

α

,wethushave

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

+ ˜

E₊

: (

T

− ˜

τ

N

)

+ (

α

− ˜

α

)

detE

˜

₊

= ˜

E₊

:

T

+

α

detE

˜

₊

−

W

( ˜

E₊

)

(33) Theright-handsideof(33)isboundedfromaboveby

(

W

−

α

det

)

∗

(

T

)

,hence

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

+ ˜

E₊

: (

T

− ˜

τ

N

)

+ (

α

− ˜

α

)

detE

˜

₊

≤ (

W

−

α

det

)

∗

(

T

)

(34) ThesameargumentbutreplacingE₊withE₋gives

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

+ ˜

E₋

: (

T

− ˜

τ

N

)

+ (

α

− ˜

α

)

detE

˜

₋

≤ (

W

−

α

det

)

∗

(

T

)

(35) Summing(34)and(35)andfurthernotingthat 1₂

( ˜

E₊

+ ˜

E₋

)

=

e_c¯N anddet E

˜

_±

= −¯

e2

_/

_{c, weobtain}

(

W

− ˜

α

det

)

∗

(

τ

˜

N

)

+

¯

e cN

: (

T

− ˜

τ

N

)

−

¯

e2 c

(

α

− ˜

α

)

≤ (

W

−

α

det

)

∗

₍

_T

₎

which,inviewof(31),canbewrittenas

(

e

¯

cN

,

−

¯

e2 c

)

∈ ∂(

W

− ˜

α

det

)

∗

₍

_τ

_˜

_N

₎

Hence

(

τ

˜

N

,

α

˜

)

satisfiestheoptimalityconditions(30),meaningthattheoptimizedbound w_

(

e

¯

)

isequalto

w



(

e

¯

;

τ

˜

N

,

α

˜

)

. Since w_

(

e

¯

;

τ

˜

N

,

α

˜

)

=

wTW

(

e

¯

)

,theconclusionisthat thebound w obtainedfromthe translationmethod coincideswith theTalbot–Willisbound wTW.

4. Example in the not strongly convex case

In thissection, we investigate an examplein which theenergy w in thematrix is not stronglyconvex, i.e. w is not convexin

e

2_{.Moreprecisely,wetake}

_{w as}

w

(

e

)

=



e if e

<

1

+∞

if e

≥

1 (36)

where

>

0 is a fixed parameter.The composite considered isa two-dimensional version of that considered in[15] for comparingtheTalbot–WillisandthePonteCastañedamethods.

TheTalbot–Willisbound wTW isgivenby

wTW

(

e

¯

)

=

⎧

⎪

⎨

⎪

⎩

e

¯

ife

¯

≤

₂₋c_c

2−c c



e

¯

2 _if c 2−c

≤ ¯

e

≤

c √ 2−c

+∞

if√c 2−c

<

e

¯

(37)

Fig. 1. Chart of the function(τ,α)→ (W−αdet)∗(τN).

ThePonteCastañedabound wPCisgivenby

wPC

(

e

¯

)

=



_2−c c



¯

e2 ife

¯

≤

√c 2−c

+∞

if√c 2−c

<

e

¯

(38) Theexpressions (37)and(38)can beobtainedbyadapting thecalculationspresentedin[15]to the2Dcase. Wenow proceedtoevaluate theboundsprovided by(12)and(13).AsinSect.3,themaindifficultyliesinsolvingthenonconvex maximizationproblem(15)thatdefinestheLegendretransform

(

W

−

α

det

)

∗

(

T

)

.Fortheenergyw in (36),the maximiza-tion problem(15)can be solved in closed form. The resultingexpression of

(

W

−

α

det

)

∗

(

τ

N

)

dependson thevalue of

(

τ

,

α

)

asrepresentedinFig. 1.Inmoredetail,wehave

(

W

−

α

det

)

∗

(

τ

N

)

=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

0 in

(

I

)

τ

−

in

(

II

)

2

τ

−

α

−

2

in

(

III

)

τ

2 2

α

+

α

−

2

in

(

IV

)

(39)

In (39), (I)–(IV)are domains in the

(

τ

,

α

)

plane that are represented in Fig. 1. The common boundary betweenthe domains(II)and(IV)istheellipticalarcwithequation

τ

2

₋

₂

_ατ

₊

₂

_α

2

₋

₂

_α

₌

_{0.Theboundarybetweenthedomains(I)}

and(IV)istheellipticalarcwithequation

τ

2

₊

₂

_α

₍

_α

₋

₂

₎

₌

_0.

Theresult(39)allowstheboundsin(12)and(13)tobeevaluated.Asafirstillustration,considerthespecialcase

τ

=

2

,

α

=

.From(13)and(39)wehave

w_

(

e

¯

;

2

N

,

)

=

2

(

4e

¯

− ¯

e

2

₋

_c

₎

₍₄₀₎

Itcaneasilybeverifiedthate

¯

<

1₂

(

4

¯

e

− ¯

e2

₋

_c

₎

_if1

₋

√

₁

₋

_c

_<

_e.

_¯

_Hence

_w



(

e

¯

;

2

N

,

)

improvesontheTalbot–Willisbound wTW foranye between

¯

1

−

√

1

−

c and c 2−c.

Better boundsthan(40)can be obtainedbyfullyoptimizing(13) withrespectto

τ

and

α

. Omittingthe detailofthe calculations,theoptimalvaluesof

(

τ

,

α

)

arefoundtodependone as

¯

follows:

– fore

¯

≤

1

−

√

1

−

c, theoptimalvaluesof

(

τ

,

α

)

areequalto

(

1

,

0

)

; – for1

−

√

1

−

c

≤ ¯

e

≤

√c

2−c,theoptimalvaluesof

(

τ

,

α

)

aregivenby

α

=

1

−

¯

e2

−

2e

¯

+

c



(

e

¯

2

₋

_2e

_¯

₊

_c

₎

2

_{+ (}

_2e

_¯

₋

_c

₎

2

,

τ

=

α

+



α

(

2

−

α

)

Forthosevalues,

(

τ

,

α

)

ison

(

II

)

∩ (

IV

)

.Thecorrespondingvalueof w_

(

¯

e

;

τ

N

,

α

)

is

w_

(

e

¯

;

τ

N

,

α

)

=



¯

e

−

1 2e

¯

2

₊

1 2



(

e

¯

2

₋

_2e

_¯

₊

_c

₎

2

_{+ (}

_2e

_¯

₋

_c

₎

2



– for √c

2−c

<

e

¯

≤

c, the optimalboundisobtainedby taking

τ

=

2

α

e

¯

/

c. For

α

large enough,thepoint

(

τ

,

α

)

isinthe

360 M. Peigney / C. R. Mecanique 345 (2017) 353–361

Fig. 2. Comparison of the bounds in the case c=0.8.

w_

(

e

¯

;

τ

N

,

α

)

=

1 2

α



¯

e22

−

c c

−

c



+

c (41)

Takingthelimit

α

→ ∞

in(41)gives

w



(

e

¯

;

τ

N

,

α

)

→ +∞

;

– for

c

<

e,

¯

theoptimalboundisobtainedbytaking

α

=

0.For

τ

largeenough,thepoint

(

τ

,

α

)

isinthedomain(III)so that

w_

(

e

¯

;

τ

N

,

0

)

=

τ

(

e

¯

−

c

)

+

c (42) Takingthelimit

τ

→ ∞

in(42)gives

w



(

e

¯

;

τ

N

,

0

)

→ +∞

.

Insummary,theoptimizedbound

w

isgivenby

w_

(

e

¯

)

=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

e

¯

ife

¯

≤

1

−

√

1

−

c



¯

e

−

1 2e

¯

2

₊

1 2



(

e

¯

2

₋

_2e

_¯

₊

_c

₎

2

_{+ (}

_2e

_¯

₋

_c

₎

2



if 1

−

√

1

−

c

≤ ¯

e

≤

√

c 2

−

c

+∞

if

√

c 2

−

c

<

e

¯

(43)

The bounds wTW, wPC and w are plotted in Fig. 2 as a function of the norm e of

¯

the effectiveelectric field. The volume fraction c of the matrixis setto 0

.

8.Fore

¯

≤

1

−

√

1

−

c, thebounds wTW and w coincidewith theenergy w

of thematrix.Themostimportantobservationisthat thetranslationbound w_ strictlyimproves bothon wTW and wPC

for 1

−

√

1

−

c

≤ ¯

e

≤

c

/

√

2

−

c. In thelimit

¯

e

→

c

/

√

2

−

c, allthe threeboundsconvergetowardsthe samevalue

c.

For

¯

e

>

c

/

√

2

−

c, thethreeboundsareequalto

+∞

. 5. Concluding remarks

Fortwo-phasecompositeswithperfectlyconductinginclusionsandastronglyconvexenergyfunction,theanalysis pre-sentedinSect.3showsthatthetranslationmethodgivesthesameHashin–Shtrikman-typeboundsastheTalbot–Willisand thePonteCastañedamethods.Itshouldbeobserved,however,thatthecalculationsinvolvedinthetranslationmethodare significantlymorecomplicatedthanthoseinvolvedintheTalbot–WillisandthePonteCastañedaprocedures.Thisismainly a consequence ofthe embedding ofthe original problemin an extended problemofdimension 4, whichis an essential ingredientofthetranslationmethodwhenappliedtotwo-dimensionalconductors.

Although itisratherspecial,the examplepresentedinSect.4showsthat thetranslationmethodhas thepotential to producenonlinearHashin–Shtrikman-typeboundsthatarestrictlystrongerthanthoseprovidedbytheTalbot–Willisandthe PonteCastañedamethods.Inthatregard,itcanbenotedthatthetranslationmethodcanbeusedinamoregeneralfashion byembeddingtheoriginalprobleminanextendedproblemofdimensionhigherthan4(byconsideringthree–ormore– copiesoftheoriginalproblem).Althoughthecalculationsbecomeevenmoreinvolved,theresultingHashin–Shtrikman-type boundsmayimproveontheTalbot–WillisandthePonteCastañedaboundsevenforstronglyconvexenergyfunctions[21].

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