Symmetry classes for even-order tensors

35 Pages·2013·0.78 MB·English

NISSUNAUMANAINVESTIGAZIONE SI PUO DIMANDARE VERASCIENZIA
S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI
LEONARDO DAVINCI
vol. 1 no. 2 2013
Mathematics and Mechanics
of
Complex Systems
MARC OLIVE AND NICOLAS AUFFRAY
SYMMETRY CLASSES FOR EVEN-ORDER TENSORS
msp
MATHEMATICSANDMECHANICSOFCOMPLEXSYSTEMS
Vol.1,No.2,2013
M ∩ M
dx.doi.org/10.2140/memocs.2013.1.177
SYMMETRY CLASSES FOR EVEN-ORDER TENSORS
MARC OLIVE AND NICOLAS AUFFRAY
We give a complete general answer to the problem, recurrent in continuum
mechanics, ofdeterminingofthenumberandtypeofsymmetryclassesofan
even-ordertensorspace. Thiskindofinvestigationwasinitiatedforthespaceof
elasticitytensors,andsincethendifferentauthorshavesolvedthisproblemfor
otherkindsofphysics,suchasphotoelectricity,piezoelectricity,flexoelectricity,
andstrain-gradientelasticity. Alltheseproblemsweretreatedusingthesame
computational method, which, though effective, has the drawback of not pro-
vidinggeneralresults. Furthermore,itscomplexityincreaseswiththetensorial
order. Hereweprovidegeneraltheoremsthatdirectlygivethedesiredresults
foranyeven-orderconstitutivetensor. Asanillustrationofthismethod,andfor
thefirsttime,thesymmetryclassesofalleven-ordertensorsofMindlinsecond
strain-gradientelasticityareprovided.
1. Introduction
Physicalmotivation. In the last years there has been increased interest in gener-
alized continuum theories; see, for example, [Forest 1998; dell’Isola et al. 2009;
2012;Lebéeand Sab2011]. Theseworks,basedonthepioneeringarticles[Toupin
1962; Mindlin1964; 1965], proposeextended kinematicformulations, to take into
account size effects within the continuum. The price to be paid for this is the
appearanceoftensorsofordergreaterthanfourintheconstitutiverelations. These
higher-order objects are difficult to handle and extracting physically meaningful
informationfromthemisnotstraightforward. Theaimofthispaperistoprovide
general results concerning the type and number of anisotropic systems an even-
ordertensorcanhave.
Suchresultshaveimportantapplications,atleast, forthemodelingandnumeri-
calimplementationofnonclassicallinearconstitutivelaws:
Modeling: The purpose of modeling is, given a material and a set of physical
variables of interest, to construct themore general constitutivelaw(a linear
one, in the present context) that describes the behavior of that material. An
exampleofsuchamethodisprovidedin[ThionnetandMartin2006],where,
MSC2010: 15A72,20C35,74B99.
Keywords: anisotropy,symmetryclasses,higher-ordertensors,generalizedcontinuumtheories,
strain-gradientelasticity.
177
178 MARCOLIVEANDNICOLASAUFFRAY
given a set of variables V and the material symmetry group S, the authors
derivemechanical behaviorlawsusingthe theory ofinvariantsandcontinuum
thermodynamics. In such regard our results will say, without making any
computationwhetherornot S iscontainedinthesetofsymmetryclassesof
L(v,v(cid:48))thespaceoflinearmapsfromv∈V tov(cid:48)∈V.
Numerical implementation: To implement a new linear constitutive law in a
finite-element code, one has to know the complete set of matrices needed
to model the associated anisotropic behavior. In that regard, our result is
a precious guideline, as it tells you how many matrices there are and how
to construct them. This is illustrated in the case of three-dimensional strain
gradientelasticityin[Auffrayetal.2013].
Constitutivetensorssymmetryclasses. Inmechanics,constitutivelawsareusually
expressedintermsoftensorialrelationsbetweenthegradientsofprimaryvariables
andtheirfluxes[GuandHe2011]. Asiswellknown,thisfeatureisnotrestricted
to linear behaviors, since tensorial relations appear in the tangential formulation
of nonlinear ones [Triantafyllidis and Bardenhagen 1996]. It is also known that
ageneraltensorialrelationcanbedividedintoclassesaccordingtoitssymmetry
properties. Suchclassesareknowninmechanicsassymmetryclasses[Forteand
Vianello 1996], and in mathematical physics as isotropic classes or strata [Abud
andSartori1983;Auffrayetal.2011].
In the case of second-order tensors, the determination of symmetry classes is
rather simple. Using spectral analysis it can be concluded that any second-order
symmetric tensor1 can either be orthotropic ([D ]), transverse isotropic ([O(2)]),
2
orisotropic ([SO(3)]). Suchtensorsare commonlyusedto describe,forexample,
heatconductionandelectricpermittivity.
For higher-order tensors, the determination of the set of symmetry classes is
moreinvolved,andismostlybasedonanapproachintroducedin[ForteandVianello
1996]forthecaseofelasticity. Letusbrieflydetailthiscase.
Thevectorspaceofelasticitytensors, denotedby(cid:69)lathroughoutthispaper,is
thesubspaceoffourth-ordertensorsendowedwiththefollowingindexsymmetries:
Minorsymmetries: E = E = E .
ijkl jikl jilk
Majorsymmetry: E = E .
ijkl klij
Symmetrieswillbespecifiedusing notation suchas E(ij)(kl),where(..)indi-
catesinvarianceunderpermutationoftheindicesinparentheses,and....indicates
invariancewithrespecttopermutationsoftheunderlinedblocks. Indexsymmetries
encodethephysicsdescribedbythemathematicaloperator. Theminorsymmetries
1Suchatensorisrelatedtoasymmetricmatrix,whichcanbediagonalizedinanorthogonalbasis.
Thestatedresultisrelatedtothisdiagonalization.
SYMMETRYCLASSESFOREVEN-ORDERTENSORS 179
stem from thefact that rigid body motionsdo not induce deformation (the symme-
try of ε), and that the material is not subjected to volumic couple (the symmetry
of σ). The major symmetry is the consequence of the existence of free energy.
Anelasticitytensor,E,canbeviewedasasymmetriclinearoperatoron(cid:84)(ij),the
spaceofsymmetricsecond-ordertensors. Accordingto[ForteandVianello1996],
for the classical action of SO(3), (cid:69)la is divided into eight symmetry classes (see
page183forthenotation):
[(cid:69)la]={[1],[Z ],[D ],[D ],[D ],[O(2)],[O],[SO(3)]},
2 2 3 4
whichcorrespond,respectively, tothefollowingphysical classes:2 triclinic,mon-
oclinic,orthotropic,trigonal,tetragonal,transverseisotropic,cubic,andisotropic.
Besidesthisfundamentalresult,theinterestoftheForteandVianellopaperwasto
provideageneralmethodtodeterminethesymmetryclassesofanytensorspace
[Auffrayetal.2011]. Otherresultshavebeenobtainedbythismethodsincethen:
Number
Property Tensor ofclasses Action Studiedin
Photoelasticity T(ij)(kl) 12 SO(3) [ForteandVianello1997]
Piezoelectricity T(ij)k 15 O(3) [GeymonatandWeller2002]
Flexoelectricity T(ij)kl 12 SO(3) [LeQuangandHe2011]
Asetoftensors
ofordersix Tijklmn 14or17 SO(3) [LeQuangetal.2012]
ThelimitationsoftheForte–Vianelloapproach. ThemethodintroducedbyForte
andVianelloisactually themostgeneral.3 But,atthesametime, itsuffersfromat
leasttwolimitations:
(1) Thecomputationoftheharmonicdecomposition.
(2) Thespecificityofthestudyforeachkindoftensor.
Initsoriginalsetting,themethodrequiresthecomputationoftheexplicithar-
monicdecompositionofthestudiedtensor,thatis,itsdecompositionintothesum
ofitsSO(3)-irreduciblecomponents,alsoknownasharmonictensors4. Itsexplicit
computation, which is generally based on an algorithm introduced by Spencer
[1970],turnsouttobeintractableinpracticeasthetensorialorderincreases. But
2ThesesymmetryclassesaresubgroupsofthegroupSO(3)ofspacerotations. Thisisbecause
theelasticitytensorisofevenorder.Totreatodd-ordertensors,thefullorthogonalgroupO(3)has
tobeconsidered.
3Someothermethodscanbefoundintheliterature,suchascountingthesymmetryplanes[Chad-
wick et al. 2001], or studying the SU(2)-action on (cid:69)la [Bóna et al. 2004], and others, but these
methodsaredifficulttogeneralizetoarbitraryvectorspaces.
4Harmonictensorsarecompletelysymmetricandtraceless.Theyinheritthisnamebecauseofa
well-knownisomorphismin(cid:82)3betweenthesetensorsandharmonicpolynomials[Backus1970].
180 MARCOLIVEANDNICOLASAUFFRAY
this is not a real problem, since the only information needed is the number of
different harmonic tensors of each order appearing in the decomposition, that is,
the isotypic decomposition. Based on arguments presented in [Jerphagnon et al.
1978], there exists a direct procedure to obtain this isotypic decomposition from
thetensorindexsymmetries[Auffray2008]. Suchanapproachhasbeenusedin
[LeQuangetal.2012]toobtainthesymmetryclassesofsixth-ordertensors.
As each kind of tensor space requires specific study, this specificity constitutes
the other limitation of the method. This remark has to be considered together
withtheobservationthat,foreven-ordertensorsitseemsthatthereexistonlytwo
possibilities. Namely,atensorspacehasasmanyclassesas
• the fullsymmetric tensorspace (forexample, (cid:69)lais divided intoeight classes,
likethefullsymmetrictensorspace[ForteandVianello1996]),or
• the generic tensor space5 (other fourth-order tensor spaces such as those of
photoelasticity[ForteandVianello1997],flexoelectricity[LeQuangandHe
2011],etc.,aredividedinto12classes,likethegenerictensorspace).
The same observation can also be made for second and sixth-order tensors
[Le Quang et al. 2012]. Understanding the general rule behind this observation
would be an important result in mechanics. Its practical implication is the direct
determination of the number and the type of symmetry classes for any constitu-
tivelaw,nomatteritsorder. Thisresultisvaluableforunderstandinggeneralized
continuumtheories,inwhichhigher-ordertensorsareinvolvedinconstitutivelaws.
Organizationofthepaper. InSection2,themainresultsofthispaper,TheoremsI,
II,andIII,arestated. Asanapplication,thesymmetryclassesoftheeven-ordercon-
stitutive tensorspacesof Mindlinsecond strain-gradientelasticity aredetermined.
Results concerning the sixth-order coupling tensor and the eighth-order second
strain-gradienttensoraregivenforthefirsttime. Obtainingthesameresultswith
theForte–Vianelloapproachwouldhavebeenmuchmoredifficult. Othersections
are dedicated to the construction of our proofs. In Section 3, the mathematical
frameworkusedtoobtainourresultisintroduced. Thereafter,westudythesym-
metry classes of a couple of harmonic tensors, which is the main purpose of the
toolnamedtheclipsoperator. Wethengivetheassociatedresultsforcouplesof
SO(3)-closedsubgroups(Theorem4.6andTable2). Thankstotheseresults,and
withthehelpofpreviousworkonthetopic[IhrigandGolubitsky1984],weobtain
inSection5somegeneralresultsconcerningsymmetryclassesforgeneraleven-
order tensors. In Section 6 our main results are finally proved. The Appendix is
devotedtoproofsandthecalculusofclipsoperations.
5Then-thordergenerictensorisan-thordertensorwithnoindexsymmetry.
SYMMETRYCLASSESFOREVEN-ORDERTENSORS 181
2. Mainresults
Inthissection,ourmainresultsarestated. Inthefirstsubsection,theconstruction
of constitutive tensor spaces (CTS) is discussed. This construction allows us to
formulateourmainresultsinthenextsubsection. Finally,theapplicationofthese
resultstoMindlinsecondstrain-gradientelasticity(SSGE)isconsidered. Precise
mathematicaldefinitionsofthesymmetryclassesaregiveninSection3.
ConstructionofCTS. Linearconstitutivelawsarelinearmapsbetweenthegradi-
entsof primaryphysical quantitiesandtheir fluxes. Eachofthese physicalquanti-
ties(seeTable1onthenextpage)isinfactrelatedtosubspaces6 oftensorsspaces;
these subspaces will be called state tensor spaces (STS). These STS will be the
primitivenotionfromwhichtheCTSwillbeconstructed.
Notation. L(F,G)willindicatethevectorspaceoflinearmapsfrom F to G.
Now we consider two STS, (cid:69) = (cid:84) and (cid:69) = (cid:84) , respectively of order p
1 G 2 f
and order q, possibly with index symmetries. As a consequence, they belong to
subspacesof(cid:78)p(cid:82)3 and(cid:78)q(cid:82)3. Aconstitutivetensor C isalinearmapbetween
(cid:69) and (cid:69) , that is, an element of the space L((cid:69) ,(cid:69) ). This space is isomorphic,
1 2 1 2
modulo the use of an euclidean metric, to (cid:69) ⊗(cid:69) . Physical properties lead to
1 2
some index symmetriesonC ∈(cid:69) ⊗(cid:69) ; thus the vector spaceof suchC is some
1 2
vectorsubspace(cid:84) of(cid:69) ⊗(cid:69) .
C 1 2
Now, each of the spaces (cid:69) , (cid:69) , and (cid:69) ⊗(cid:69) has natural O(3) actions. In this
1 2 1 2
paper,weareconcernedwithcasesinwhich p+q =2n. Insuchasituation,itis
knownthattheO(3)-actionon(cid:69) ⊗(cid:69) reducestothatofSO(3)[ForteandVianello
1 2
1996]. Wethereforehave
L((cid:69) ,(cid:69) )(cid:39)(cid:69) ⊗(cid:69) ⊂(cid:84)p⊗(cid:84)q =(cid:84)p+q=2n.
1 2 1 2
Herearesomeexamplesofthisconstruction:
Property (cid:69) (cid:69) Tensorproduct Number
1 2
forCTS ofclasses
Elasticity (cid:84)(ij) (cid:84)(ij) Symmetric 8
Photoelasticity (cid:84)(ij) (cid:84)(ij) Standard 12
Flexoelectricity (cid:84)(ij)k (cid:84)i Standard 12
First-gradientelasticity (cid:84)(ij)k (cid:84)(ij)k Symmetric 17
ThistableshowstwokindsofCTS,describingrespectively:
• coupledphysics(tensorssuchasphotoelasticityandflexoelectricity,encoding
thecouplingbetweentwodifferentphysics),and
6Becauseofsomesymmetries.
182 MARCOLIVEANDNICOLASAUFFRAY
Physicalnotion Mathematicalobject Mathematicalspace
Gradient TensorstateT ∈(cid:78)p(cid:82)3 (cid:84) : tensorspacewith
1 G
indexsymmetries
Fluxesofgradient TensorstateT ∈(cid:78)q(cid:82)3 (cid:84) : tensorspacewith
2 f
indexsymmetries
Linearconstitutivelaw C ∈L((cid:84) ,(cid:84) ) (cid:84) ⊂L((cid:84) ,(cid:84) )
G f C G f
Table1. Physicalandmathematicallinks.
• properphysics(tensorssuchasclassicalandfirst-gradientelasticities,describ-
ingasinglephysicalphenomenon).
Onthemathematicalsidethisimplies:
• Coupled physics: the spaces (cid:69) and (cid:69) may differ, and when (cid:69) =(cid:69) linear
1 2 1 2
mapsarenotself-adjoint.
• Properphysics: wehave(cid:69) =(cid:69) andlinearmapsareself-adjoint.7
1 2
Therefore,theelasticitytensor isaself-adjointlinearmapbetweenthevector
space of deformation tensors and the vector space of stress tensors. These two
spaces are modeled on (cid:84)(ij). The vector space of elasticity tensors is therefore
completely determined by (cid:84)(ij) and the symmetric nature of the tensor product,
that is, (cid:69)la=(cid:84)(ij)⊗S(cid:84)(kl), where ⊗S denotes the symmetric tensor product. On
the side of coupling tensors, flexoelectricity is a linear map between (cid:69)1 =(cid:84)(ij)k,
thespaceofdeformationgradients,and(cid:69) =(cid:84) ,theelectricpolarization;therefore
2 l
(cid:70)lex=(cid:84)(ij)k⊗(cid:84)l.
Symmetryclassesofeven-ordertensorspaces. Consideraneven-orderCTS(cid:84)2n.
It is known [Jerphagnon et al. 1978] that thisspace can be decomposed orthogo-
nally8 intoafullsymmetricspaceandacomplementaryonewhichisisomorphic
toatensorspaceoforder2n−1,thatis:
(cid:84)2n =(cid:83)2n⊕(cid:67)2n−1.
Letusintroduce:
(cid:83)2n: thevectorspaceof2n-thordercompletelysymmetrictensors.
(cid:71)2n: thevectorspaceof2n-thordertensorswithnoindexsymmetries.9
Thefollowingobservationisobvious:
(cid:83)2n ⊆(cid:84)2n ⊆(cid:71)2n,
7Thisisaconsequenceoftheassumptionoftheexistenceofafreeenergy.
8Therelateddotproductisconstructedby2nproductsofthe(cid:82)3canonicalone.
9Formallythisspaceisconstructedas(cid:71)2n=(cid:78)2n(cid:82)3.
SYMMETRYCLASSESFOREVEN-ORDERTENSORS 183
andtherefore,ifwedenotebyItheoperatorwhichgivestoatensorspacetheset
ofitssymmetryclasses,weobtain:
I((cid:83)2n)⊆I((cid:84)2n)⊆I((cid:71)2n).
Thesymmetrygroupofeven-ordertensorsisconjugatetoSO(3)-closedsubgroups
[ZhengandBoehler1994;ForteandVianello1996]. ClassificationofSO(3)-closed
subgroups is a classical result that can be found in many references [Ihrig and
Golubitsky1984;Sternberg1994]. Thesesubgroupsare,uptoconjugacy:
Lemma2.1. Everyclosedsubgroupof SO(3)isconjugatetopreciselyonegroup
fromthefollowinglist:
{1,Z ,D ,T,O,I,SO(2),O(2),SO(3)}.
n n
Amongthesegroups,wecandistinguish:
Planargroups: {1,Z ,D ,SO(2),O(2)},whichareO(2)-closedsubgroups.
n n
Exceptionalgroups: {T,O,I,SO(3)},ofwhichthefirstthreearerotationalsym-
metrygroupsofPlatonicpolyhedra.
Letusdetailfirstthesetofplanarsubgroups. Wefixabase(i;j;k)of(cid:82)3,and
denotebyQ(v;θ)∈SO(3)therotationaboutv∈(cid:82)3,withangleθ ∈[0;2π).
• 1istheidentity.
• Z (n ≥ 2) is the cyclic group of order n, generated by the n-fold rotation
n
Q(k;θ =2π/n). whichisthesymmetrygroupofachiralpolygon.
• D (n ≥ 2) is the dihedral group of order 2n generated by Z and Q(i;π),
n n
whichisthesymmetrygroupofaregularpolygon.
• SO(2)isthesubgroupofrotationsQ(k;θ)withθ ∈[0;2π).
• O(2)isthesubgroupgeneratedbySO(2)andQ(i;π).
The classes of exceptional subgroups are: T, the tetrahedral group of order
12 which fixes a tetrahedron, O, the octahedral group of order 24 which fixes an
octahedron(oracube),andI,thesubgroupoforder60whichfixesanicosahedron
(oradodecahedron).
InSection6,thesymmetryclassesof(cid:83)2n and(cid:71)2n areobtained:
Lemma2.2. Thesymmetryclassesof (cid:83)2n are:
I((cid:83)2)={[D ],[O(2)],[SO(3)]},
2
I((cid:83)4)={[1],[Z ],[D ],[D ],[D ],[O(2)],[O],[SO(3)]},
2 2 3 4
I((cid:83)2n)={[1],[Z2],...,[Z2(n−1)],[D2],...,[D2n],[O(2)],[T],[O],[I],[SO(3)]},
ifn≥3.
184 MARCOLIVEANDNICOLASAUFFRAY
Lemma2.3. Thesymmetryclassesof (cid:71)2n are:
I((cid:71)2)={[1],[Z ],[D ],[SO(2)],[O(2)],[SO(3)]},
2 2
I((cid:71)4)={[1],[Z ],[Z ],[Z ],[D ],[D ],[D ],[SO(2)],[O(2)],[T],[O],[SO(3)]},
2 3 4 2 3 4
I((cid:71)2n)={[1],[Z ],...,[Z ],[D ],...,[D ],[SO(2)],[O(2)],[T],[O],[I],[SO(3)]},
2 2n 2 2n
ifn≥3.
Thefollowingtablelistshowmanyclassesthereareforeachn:
n= 1 2 ≥3
#I((cid:83)2n) 3 8 2(2n+1)
#I((cid:71)2n) 6 12 4n+5
Thesymmetryclassesof(cid:84)2n areclarifiedbythefollowingtheorem:
TheoremI. Let(cid:84)2n beatensorspace. TheneitherI((cid:84)2n)=I((cid:83)2n)orI((cid:84)2n)=
I((cid:71)2n).
Inotherwords,thenumberandtypeofclassesarethesameasthoseof
• (cid:83)2n,thespaceof2n-ordercompletelysymmetrictensors,inwhichcasethe
numberofclassesisminimal,or
• (cid:71)2n,thespaceof2n-ordergenerictensors,inwhichcasethenumberofclasses
ismaximal.
Infact,asspecifiedbythefollowingtheorems,inmostsituationsthenumberof
classesisindeedmaximal.
TheoremII(couplingtensors). Letusconsider(cid:84)2p thespaceofcouplingtensors
between two physics described by two tensor vector spaces (cid:69) and (cid:69) . If these
1 2
tensorspacesareofordersgreaterthanorequaltoone,thenI((cid:84)2p)=I((cid:71)2p).
TheoremIII(propertensors). Letusconsider(cid:84)2p,thespaceoftensorsofaproper
physicsdescribedbythetensorvectorspace(cid:69). Ifthistensorspaceisoforder p≥3,
andissolelydefinedintermsofitsindexsymmetries,thenI((cid:84)2p)=I((cid:71)2p).
Remark2.4. Exceptionsoccurfor
p=1, whenthespaceofsymmetricsecond-ordertensorsisobtained,and
p=2, when,inthecaseof(cid:84)(ij),thespaceofelasticitytensorsisobtained.
Ineachofthesesituationsthenumberofclassesisminimal. Thereisnoother
situationwherethis caseoccurs. Itshould thereforebeconcludedthatthe spaceof
elasticitytensorsisexceptional.
SYMMETRYCLASSESFOREVEN-ORDERTENSORS 185
Secondstrain-gradientelasticity(SSGE). Applicationoftheformertheoremswill
bemadeontheeven-ordertensorsofSSGE.First,theconstitutiveequationswill
be summed up, and then the results will be stated. It worth noting that obtain-
ingthe sameresultswith theForte–Vianelloapproach wouldhavebeenfar more
complicated.
Constitutivelaws. Inthesecondstrain-gradienttheoryoflinearelasticity[Mindlin
1965;Forestetal.2011],the constitutivelawgivesthesymmetricCauchystress
tensor10 σ(2) and thehyperstress tensorsτ(3) andω(4) in termsof the infinitesimal
straintensorε(2) anditsgradientsη(3)=ε(2)⊗∇ andκ(4)=ε(2)⊗∇⊗∇ through
thethreelinearrelations:
σ(2)=E(4):ε(2)+M(5)∴η(3)+N(6)::κ(4),
τ(3)=MT(5):ε+A(6)∴η(3)+O(7)::κ(4), (2-1)
ω(4)=NT(6):ε(2)+OT(7)∴η(3)+B(8)::κ(4),
where :, ∴, and :: denote, respectively, the double, third, and fourth contracted
products. Above,11 σ(ij),ε(ij),τ(ij)k,η(ij)k =ε(ij),k,ω(ij)(kl),andκ(ij)(kl)=ε(ij),(kl)
are,respectively,thematrixcomponentsofσ(2),ε(2),τ(3),η(3),ω(4),andκ(4) rela-
tivetoanorthonormalbasis(i;j;k)of(cid:82)3. And E(ij)(lm), M(ij)(lm)n, N(ij)(kl)(mn),
A(ij)k (lm)n, O(ij)k(lm)(no), and B(ij)(kl)(mn)(op) are the matrix components of the
relatedelasticstiffnesstensors.
Symmetryclasses. Thesymmetryclassesoftheelasticitytensorsandofthefirst
strain-gradient elasticity tensors has been studied in [Forte and Vianello 1996;
Le Quang et al. 2012]. Hence, here we solely consider the spaces of coupling
tensorsN(6) andofsecondstrain-gradientelasticitytensorsB(8).
• Wedefine(cid:67)estobethespaceofcouplingtensorsbetweenclassicalelasticity
andsecondstrain-gradientelasticity:
(cid:67)es={N(6)∈(cid:71)6| N(ij)(kl)(mn)}.
AdirectapplicationofTheoremIIleadstotheresultthat
I((cid:67)es)={[1],[Z ],...,[Z ],[D ],...,[D ],[SO(2)],[O(2)],[T],[O],[I],[SO(3)]}.
2 6 2 6
Therefore(cid:67)esisdividedinto17symmetryclasses.
• Wedefine(cid:83)grtobethespaceofsecondstrain-gradientelasticitytensors:
(cid:83)gr={O(8)∈(cid:71)8| O(ij)(kl)(mn)(op)}.
10Inthissubsectiononly,tensororderswillbeindicatedbysuperscriptsinparentheses.
11Thecommaclassicallyindicatesthepartialderivativewithrespecttospatialcoordinates.Su-
perscriptT denotestransposition.Transpositionisdefinedbypermutingthe pfirstindiceswiththe
qlast,where pisthetensorialorderoftheimageofaq-ordertensor.