Depósi o de In es igación de la Uni e sidad de Se illa
h ps://idus.us.es/
This is an Accep ed Manusc ip o an a icle published by Else ie
in Enginee ing Analysis wi h Bounda y Elemen s, Volume 105, on Augus
2019, a ailable a : h ps://doi.o g/10.1016/j.enganabound.2019.04.010
Copy igh 2019 Else ie . En idUS Licencia C ea i e Commons CC BY-NC-ND
A as andnon-degene a escheme
o he e alua ion o he 3D undamen al solu ion
and i s de i a i es o ully aniso opic
magne o-elec o-elas ic ma e ials✩
Fede ico C. Bu onia,∗,C is ianoUbessi
b,Gab ielHa o i
c,
Rogé io J. Ma czakb,And ésSáez
d
aDepa men o Mechanical Enginee ing and Manu ac u ing, Uni e sidad de Se illa.
Camino de los Descub imien os s/n , Se ille E-41092, Spain
bDepa men o Mechanical Enginee ing - DEMEC/PROMEC, Uni e sidade Fede al do
Rio G ande do Sul. Sa men o Lei e 425, 90050-160, Po o Aleg e,B azil
cDepa men o Enginee ing, Uni e si y o Camb idge, CB2 1PZ, Camb idge, UK
dDepa men o Con inuum Mechanics and S uc u al Analysis, Uni e sidad de Se illa.
Camino de los Descub imien os s/n , Se ille E-41092, Spain
Abs ac
Anewexp ession o he undamen alsolu ionisin oduced,p esen ing
h ee ele an cha ac e is ics: (i)i isexplici in e mso heS oh’seigen al-
ues, (ii)i emainswell-de inedwhensomeS oh’seigen aluesa e epea ed,
and (iii)i isexac . A as and obus nume icalscheme o hee alua ion o
he undamen al solu ion and i s de i a i es de eloped omdoubleFou ie
se ies ep esen a ions is p esen ed. The Fou ie se ies ep esen a ion is pos-
sible due o he pe iodic na u e o he solu ion. The a ac i eness o his
se ies solu ion is ha he in o ma ion o he ma e ial p ope ies is con ained
only in he Fou ie coefficien s, while he in o ma ion o he dependence o
he e alua ion poin is con ained in simple igonome ic unc ions. This
implies ha any o de de i a i es can be de e mined by spa ial diffe en ia-
ion o he igonome ic unc ions. Mo eo e , Fou ie coefficien s need o be
ob ained only once o a gi en ma e ial, leading o an efficien me hodology.
✩Special issue on he T ibu e o he B azilian Compu a ional Mechanics Communi y
o Ca los B ebbia: Edi ed by Ney Augus o Dumon and José Telles
∗co esponding au ho
Email add ess: [email p o ec ed] (Fede ico C. Bu oni)
P ep in submi ed o EABE Ap il 1, 2019
The obus ness o he scheme a ises om he p ope ies (i)and(ii)o he
new exp ession o he undamen al solu ion, which is used ocompu e he
Fou ie coefficien s. The p oposed app oach combines he clean s uc u e o
he S oh o malism wi h he simplici y o Fou ie expansions, add essing he
old d awbacks o aniso opic undamen al solu ions.
Key wo ds: explici exp essions, ma hema ical degene a e ma e ials,S oh
o malism, h ee-dimensional magne oelec oelas ici y,G een’s unc ions,
Fou ie se ies ep esen a ion
1. In oduc ion
In he las ew yea s, he Magne o-Elec o-Elas ic (MEE) coupling p e-
sen ed in composi es consis ing o Piezoelec ic (PE) and Piezomagne ic
(PM) phases has been he ocus o in ensi e esea ch due o henume ous
eme ging possibili ies a mul iple ma e ial scales. These coupling phenomena
ha e also used o mode n applica ions in a ious de ices: senso s, ac ua o s
and sma s uc u es a e jus some o he cu en applica ions. New possi-
bili ies in ab ica ion and design o ad anced ma e ials o gene al aniso opy
and no el mic o/nano de ices and s uc u es equi e obus and efficien
nume ical me hods o he analysis o aniso opic mul icoupling ma e ials.
His o ic de elopmen s, challenges and pe spec i es o MEE ma e ials and i s
applica ions can be ound in ecen e iews [1, 2].
Fundamen al solu ions can be used o sol e sophis ica ed bounda y alue
p oblems. They a e a cen al subjec in modelling he physical and mechan-
ical beha iou o solids. Fo ins ance, hey a e equi ed o ob ain s esses
due o in e nal de ec s in ma e ials. They a e also equi ed o he analy-
sis o gene al p oblems by some nume ical me hods, such as heBounda y
Elemen Me hod (BEM) [3–5], o which P o esso Ca los B ebbia has made
ex ensi e con ibu ions [6]. In all hese me hodologies, undamen al solu ions
a e ypically e alua ed housands o millions o imes. Ad ances in he use
o hese nume ical me hods a e limi ed by he a ailabili y o efficien and
obus nume ical schemes o he co esponding undamen alsolu ions.Ad-
di ionally, hese solu ions equi e ex ensi e compu a ions, which can ende
he nume ical me hod in easible o la ge p oblems.
All mul i ield ma e ials ha e some deg ee o aniso opy. Hence, he de-
elopmen o aniso opic undamen al solu ions wi h coupled beha iou has
been buil upon p e ious esea ch o pu ely elas ic aniso opic elas ici y his-
2
o ically. A ecen comp ehensi e e iew o his de elopmen can be ound in
Muñoz-Reja e al. [7], so we ocus ou a en ion in he main d awbacks o ex-
is ing h ee-dimensional (3D) aniso opic MEE solu ions. The main in e es
in inding undamen al solu ions in a sui able o m o hei nume ical imple-
men a ion has been he compu a ional cos . Fo compa ison pu poses, he
gene al aniso opic elas ic case con ains 21 diffe en elas ic cons an s, how-
e e he amoun o calcula ions o he undamen al solu ionisal eadyla ge.
Fo he MEE case, wi h 75 diffe en ma e ial cons an s, he compu a ional
cos becomes a key issue. In gene al, undamen al solu ions a e a ailable in
he li e a u e ei he in in eg al o m o in e ms o explici exp essions.
On one hand, undamen al solu ion and i s de i a i es may be p esen ed
in in eg al o m (see he wo k by Han [8] and e e ences he ein). This ap-
p oach is called implici and in ol es nume ical in eg a ion o egula unc-
ions. When he in eg al pa o he undamen al solu ion is a anged in
o de o be dependen only on he di ec ion o he e alua ion ec o bu
no on i s modulus (see o ins ance deduc ion by Wang & Achenbach [9]
o he elas ic case), he smoo hness o he ke nels depends on he deg ee o
aniso opy. In al e na i e in eg al o ms, co esponding ke nels may become
less smoo h and mo e difficul o in eg a e o small modulus o heposi ion
ec o [10]. A any e en , ega ding he BEM implemen a ion o he unda-
men al solu ions, he nume ical compu a ion o he in ol edin eg alsmay
lead in many cases o inefficien and compu a ionally cos ly schemes.
On he o he hand, explici exp essions a e mo e desi able, as p o ided by
Pan [11] o he undamen al solu ion o MEE ma e ials and by Bu oni and
Sáez [12] o i s de i a i es. In his app oach, one needs o sol e an eigen-
p oblem o each e alua ion poin , and one deals wi h he ypical p oblem
o ma hema ical degene acy which can a ise in explici o mula ions. Bu oni
and Sáez [12] ha e p esen ed a se o explici solu ions, bo h o he un-
damen al solu ion and i s de i a i es, which conside all possible kind o
degene acies – howe e one needs o know which solu ion o apply, and his
is no a i ial ask. In addi ion o he abo e-men ioned con ibu ions, ad-
di ional effo has been in es ed in ecen yea s o cons uc ing adequa e
explici solu ions. Fo ins ance, see he wo ks o Xie e al. [13] and [14] o
PE and MEE ma e ials, espec i ely, whe e bo h solu ions degene a e ma h-
ema ically. I is ema kable he explici app oach ecen lyp oposedbyXiee
al. [15] o PE ma e ials, which is alid o degene a e cases. Howe e , o he
au ho s’ bes knowledge no non-degene a e explici undamen al solu ion has
been de i ed o MEE ma e ials. Asolu ioncons uc edbyRadon-S oh o -
3
malism [10, 17] has been p oposed and wo diffe en e alua ionapp oaches,
complex o m and eal o m, ha e been sugges ed and discussedinHsue
al. [16]. Be ween hese wo app oaches, he one based upon hecomplex
o m also has he ea u e o single ime e alua ion, a concep simila o he
p esen ed in his s udy.
Mo eo e , some app oxima e app oaches such as he da a base and in e -
pola ion scheme p oposed by Wilson & C use [18] ha e also beenp oposed.
See o ins ance he wo k by Mu aishi [19] o PE ma e ials. Ano he in e -
es ing app oach based on he ela ionship be ween he Rayleigh expansion
and Fou ie ep esen a ion, which leads o a se ies ep esen a ion in e ms o
sphe ical ha monics has been p oposed ecen ly [20]. In he con ex o con-
ac and c ack p oblems see he wo k by Fab ikan [21] and o he mo-MEE
undamen al solu ions he wo k by Pas e nak e al. [22].
In summa y, exis ing undamen al solu ions exhibi a leas oneo he ol-
lowing incon eniences: (i) heya eimplici ,o (ii) heya ema hema ically
degene a ed, p e en ing he use o such solu ions in he gene al case, and/o
(iii) heya ee alua edbyanapp oxima edp ocedu e.In hiscon ex , he
aim o his wo k is o p o ide a nume ical scheme sui able o nume ical im-
plemen a ion o he 3D undamen al solu ion and i s i s - andsecond-o de
de i a i es o PE, PM and MEE ma e ials wi h gene al aniso opy. The
emainde o he pape is o ganized as ollows: In Sec ion 2, basic equa ions
and no a ion o magne oelec oelas ici y a e in oduced.Ino de oo e -
come he abo e men ioned limi a ions, in Sec ion 3 a new exp ession o he
undamen al solu ion in MEE ma e ials is p esen ed which has h ee ele an
cha ac e is ics: (i)i isexplici in e mso heS oh’seigen alues,(ii)i
emains well-de ined when some S oh’s eigen alues a e equal(ma hema ical
degene acy) o nea ly equal (quasi-ma hema ical degene acy), and (iii)i is
exac . Nex , by using his new exp ession and a ep esen a ion o he solu ion
based on double Fou ie se ies, a e y as and obus nume ical scheme o
he e alua ion o he 3D undamen al solu ion and i s de i a i es is p esen ed
in Sec ion 4. Some nume ical esul s alida e he p oposed app oach while
showcasing he a ained accu acy. We discuss he conclusions in Sec ion 5.
2. Basic equa ions o linea magne oelec oelas ici y
Le xi(i=1,2,3)beaCa esiancoo dina esys emin h ee-dimensions.
The ex ended no a ion in oduced by Ba ne & Lo he [23] o PE ma e ials
is e y con enien o he pu pose o his wo k. In his way, he linea MEE
4

p oblem can be o mula ed in an elas ic-like ashion by ex ending he elas ic
displacemen ield ec o uiwi h he addi ion o he elec ic po en ial ϕand
he magne ic po en ial ϑas [24]
uJ=⎧
⎨
⎩
ujJ⩽3
ϕJ=4
ϑJ=5,
(1)
and by de ining an ex ended elas ici y enso wi h he ollowing componen s
[24]
CiJKm =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
cijkm J, K ⩽3
emij J⩽3; K=4
eikm J=4; K⩽3
qmij J⩽3; K=5
qikm J=5; K⩽3
−λim J=4; K=5o J=5; K=4
−ϵim J, K =4
−µim J, K =5.
(2)
whe e cijkl,ϵil and µil deno e he componen s o he elas ic s iffness enso
a cons an elec ic and magne ic ields, he dielec ic pe mi i i y enso a
cons an s ains and magne ic ields, and he magne ic pe meabili ies enso
a cons an s esses and elec ic displacemen s, espec i ely; eijk,qijk and
λil a e he PE coupling coefficien s a cons an magne ic ields, PMcoupling
coefficien s a cons an elec ic ields and ME coupling coefficien s a cons an
s ains and elec ic ields, espec i ely. We assume iso he mal condi ions.
The ma e ial cons an s enso s show he ollowing symme y condi ions
cijkl =cjikl =cijlk =cklij,e
kij =ekji,q
kij =qkji,(3)
ϵkl =ϵlk,λkl =λlk,µ
kl =µlk.
Due o hese symme ies, CiJKm =CmKJi is sa is ied. Mo eo e , he elas ic
cons an , dielec ic pe mi i i y, magne ic pe meabili yandMEcoupling
enso s a e posi i e de ini e, i.e.
cijkmγijγkm >0,ϵijEiEj>0,µ
ijHiHj>0,λimEiHj>0(4)
∀γkm,E
j,H
j∈R;γkm =γmk =0,E
j=0,H
j=0.
5
and no cons ain is imposed on he PE and PM coupling enso s.Equa ion
(4) is known as he s ong con exi y condi ion and i is equi alen o posi i e
de ini eness o he in e nal ene gy unc ion.
In he de ini ions abo e, he lowe case (elas ic) and uppe case (ex ended)
subsc ip s ake alues 1, 2, 3 and 1, 2, 3 (elas ic), 4 (elec ic), 5 (magne ic),
espec i ely. As poin ed ou by Fan [25] o piezoelec ici y, hese ma ix
ep esen a ions a e no enso s. So one has o be ca e ul whenchangingco-
o dina es sys ems (as he ones used in Sec ion 4). Then, using hein oduced
ma ix ep esen a ion, ellip ic equilib ium equa ions o heelas ic,elec ic
and magne ic p oblems in e ms o he ex ended displacemen scanbe ecas
in a simila way o Na ie ’s equa ion o elas ici y as
CiJKmuK,mi + J=0,(5)
whe e Jis he ex ended body o ce ec o , de ined as
J=⎧
⎨
⎩
jJ⩽3
− eJ=4
− mJ=5,
(6)
being i, eand m he h ee componen s o body o ces, he elec ic cha ge
densi y and he elec ic cu en densi y, espec i ely. As usual, comma de-
no es diffe en ia ion. No e ha uncoupled p oblems, i.e.,pu elyelas ic,
elec ic and/o magne ic, can be conside ed by se ing he co esponding
coefficien s eijk,qijk and/o λil o ze o.
3. Non-degene a e undamen al solu ion
The undamen al solu ion is de ined as a wo-poin second-o de enso
in a i e-dimension space wi h componen s UKP such ha sa is ies he ellip-
ic pa ial diffe en ial equa ions (5) whe e he gene alisedbody o ce ec o
co esponds o a poin load J=δJPδ(x−x′)being δ(x−x′) he Di ac
del a unc ion loca ed a he sou ce poin x′and δJK he i e-dimension K o-
necke del a. In homogeneous media he undamen al solu ions depends on
he ela i e ec o x−x′so, o simplici y i is conside ed ha he Ca esian
coo dina e sys em has he o igin a he sou ce poin x′, hus he undamen al
solu ion is jus a unc ion o he e alua ion poin x.Fo aphysicalin e p e-
a ion see [12].
6
The undamen al solu ion can be exp essed as a singula e m by a mod-
ula ion unc ion Has
UJK(x)= 1
4π HJK(x)(7)
whe e x= ˆe wi h =|x|=0.Themodula ion unc ionHJK(x)depends
on he di ec ion o xbu no on i s modulus, so HJK(x)=HJK(ˆe ).This
unc ion can be pu in he con ex o he S oh o malism [26] being known
as one o he h ee ex ended Ba ne -Lo he enso s, which is symme ic and
H(ˆe )=H(−ˆe ).Hence,U(x)is also symme ic and e en, i.e.:
UJK(x)=UJK(−x).(8)
The e o e, he ollowing pa i y ela ionships o he de i a i es o U(x)
UJK,m(x)=−UJK,m(−x)(9)
and
UJK,mn(x)=UJK,mn(−x).(10)
a e sa is ied.
The enso HJK can be e alua ed as [12, 27]
HJK(ˆe )=1
π%+∞
−∞
Γ−1
JK(p)dp, (11)
wi h
ΓJK(p)=QJK +(RJK +RKJ)p+TJKp2,(12)
and
QJK =CiJKmninm,R
JK =CiJKmnimm,T
JK =CiJKmmimm,(13)
whe e niand mia e he componen s o any wo mu ually o hogonal uni
ec o s such ha (n,m,ˆe )is a igh -handed iad. No e ha QJK and TJK
a e symme ic like hei elas ic coun e pa s [26], bu he MEE coupling
cause he loss o posi i e de ini eness o hese ma ices. Howe e , as Ting
[26] shows o piezoelec ic ma e ials, i can be p o en ha QJK,TJK and
ΓJK a e non-singula in his case, so hei in e ses a e gua an eed. Mo eo e ,
Hand Ua e independen o he choice o he uni ec o s mand non he
oblique plane.
7
The ke nel in equa ion (11) is a single- alued holomo phic unc ion in
he uppe complex hal -plane excep a he i e complex poles wi h posi i e
imagina y pa and hei conjuga es ha co esponds o he oo s o he
en-o de polynomial equa ion
|Γ(p)|=0.(14)
The de e minan in (14) can be ac o ised as
|Γ(p)|=|T|
5
&
ξ=1
(p−pξ)(p−¯pξ),(15)
whe e pξa e known as he S oh’s eigen alues and he ba o e pξdeno es he
complex conjuga e and Tis de ined in equa ion (13). S oh’s eigen alues can
be ob ained as he oo s o en h-o de cha ac e is ic equa ion (14) as well
as sol ing he eigen-p oblem, as desc ibed in Appendix A. Then, assuming
ha all S oh’s eigen alues a e diffe en , he in eg a ion in equa ion (11) can
be done by he Cauchy’s esidue heo y o yield
HJK(ˆe )= 2i
|T|
5
'
α=1
ˆ
ΓJK(pα)
(pα−¯pα)
5
(
ξ=1
ξ=α
(pα−pξ)(pα−¯pξ)
,(16)
whe e ˆ
ΓJK is he adjuga e o ΓJK de ined as ΓPJ(p)ˆ
ΓJK(p)=|Γ(p)|δPK and
i=√−1.Clea ly, hisexp essionisno alid o degene a ecaseswhen
he e a e epea ed S oh’s eigen alues. Following an idea o [27],equa ion
(16) can be algeb aically modi ied in o de o ob ain a well-de ined solu ion,
alid e en o epea ed S oh’s eigen alues o any mul iplici y, as explained
below.
The adjuga e ma ix ˆ
ΓJK(p)is a polynomial in po deg ee eigh . Le
ˆ
ΓJK(p)=
8
'
n=0
pnˆ
Γ(n)
JK,(17)
whe e ˆ
Γ(n)
JK (n=0,...,8) a e eal symme ic ma ices which only depend
on he ma e ial p ope ies and he posi ion ec o ˆe .These ma ices can
be compu ed in a s aigh o wa d way in e ms o he S oh ma ices (13)
8
Then, poin s on he x3-axis such as ˆe (3) =(0,0,1)To ˆe (3) =(0,0,−1)T,in
he new coo dina e sys em x∗
i(i=1,2,3) heybecomeˆe ∗
(3) =(−1,0,0)To
ˆe ∗
(3) =(1,0,0)T, espec i ely. Thisa i icecanbeusedsince heexp essions
in equa ions (28) and (30) a e well-de ined o ec o s like ˆe ∗
(3).Modula ion
unc ions ˜
UIKj and ˜
UIKjl a e no enso s. They can be ans o med acco ding
o he ollowing ules
˜
UIKj(ˆe (3))=Ω(2)|π
2
AI Ω(2)|π
2
BK Ω(2)|π
2
cj ˜
U∗
ABc(ˆe ∗
(3)),(32)
and
˜
UIKjl(ˆe (3))=Ω(2)|π
2
AI Ω(2)|π
2
BK Ω(2)|π
2
cj Ω(2)|π
2
dl ˜
U∗
ABcd(ˆe ∗
(3)).(33)
Na u ally, since Ω(2)|π
2is no necessa ily a ma e ial symme y ope a ion, he
MEE p ope ies mus be exp essed in he x∗
i(i=1,2,3)sys emas
c∗
ijkl =Ω(2)|π
2
ia Ω(2)|π
2
jb Ω(2)|π
2
kc Ω(2)|π
2
ld cabcd,(34)
e∗
ijk =Ω(2)|π
2
ia Ω(2)|π
2
jb Ω(2)|π
2
kc eabc,(35)
q∗
ijk =Ω(2)|π
2
ia Ω(2)|π
2
jb Ω(2)|π
2
kc qabc,(36)
ϵ∗
ij =Ω(2)|π
2
ia Ω(2)|π
2
jb ϵab,(37)
λ∗
ij =Ω(2)|π
2
ia Ω(2)|π
2
jb λab,(38)
µ∗
ij =Ω(2)|π
2
ia Ω(2)|π
2
jb µab,(39)
in o de o compu e he solu ions in he x∗
i(i=1,2,3)coo dina esys em.
Table 2 compa es he undamen al solu ion o ully MEE Ma e ial C
(p ope ies lis ed in able B.7) a poin x=(1,1,1)Tob ained wi h he
Fou ie se ies app oach, he non-degene a e app oach and he solu ion by
Bu oni & Sáez [12] as e e ence. Table 3 p esen s he esul s ob ained wi h he
p oposed o mula ion and i s compa ison wi h he esul s by Bu oni & Sáez
and ini e diffe ence app oach o he UJK,3de i a i es a poin x=(1,1,1)T
[12]. Table 4 p esen s he compa a ison o he UJK,12 de i a i es a he
same poin wi h he ini e diffe ence app oach by using he Bu oni & Sáez
solu ion [12]. Fo all cases, 256 Gauss poin s ha e been used o in eg a ion
o he coefficien s (21), and he se ies ake 20 e ms. Ve y goodag eemen
is obse ed amongs he solu ions o all cases.
15

(JK)Bu oni & Sáez (2010) p esen wo k (non degene a e) p esen wo k (Fou ie Se ies ρ=2)
11 8.7225398002742391 ×10−48.7225310879278819 ×10−48.7225302692569269 ×10−4
12 6.4154620180000892 ×10−56.4154582776641815 ×10−56.4154583909996026 ×10−5
13 3.1446825949164037 ×10−43.1446793068922219 ×10−43.1446789927188727 ×10−4
14 −1.2680519416303164 ×10−3−1.2680506415118376 ×10−3−1.2680505211279109 ×10−3
15 −5.1231568254934725 ×10−7−5.1231513273905938 ×10−7−5.1231508865968785 ×10−7
22 7.6090426614139801 ×10−47.6090350460375387 ×10−47.6090344129206722 ×10−4
23 4.9374127465890965 ×10−54.9374095746782780 ×10−54.9374098591871851 ×10−5
24 4.5315107542192896 ×10−44.5315058718214098 ×10−44.5315053684158277 ×10−4
25 2.8343787079494569 ×10−72.8343751170138947 ×10−72.8343748212786792 ×10−7
33 1.0440331060742933 ×10−31.0440321207110419 ×10−31.0440320278586277 ×10−3
34 −1.8426611715379502 ×10−3−1.8426594309862083 ×10−3−1.8426592695127788 ×10−3
35 −6.5916283018091271 ×10−7−6.5916214947368479 ×10−7−6.5916209154751710 ×10−7
44 1.2157849166703620 ×10−21.2157837419870272 ×10−21.2157836354733913 ×10−2
45 1.8466901349619211 ×10−61.8466881829979327 ×10−61.8466880294716113 ×10−6
55 2.5182299673182738 ×10−72.5182273648575246 ×10−72.5182271523844513 ×10−7
Table 2: G een’s unc ion UJK a e alua ion poin x=(1,1,1)T o MEE Ma e ial C.
(JK)Fini e diffe ence Bu oni & Sáez (2010) p esen wo k (Fou ie Se ies ρ=2)
11 −3.675418036705983 ×10−13 −3.675418038147529 ×10−13 −3.6754151848859640 ×10−13
12 −1.022473809428751 ×10−13 −1.022473811350401 ×10−13 −1.0224716811402825 ×10−13
13 −1.130914219850401 ×10−14 −1.130914522684852 ×10−14 −1.1309163083534299 ×10−14
14 3.207066889615237 ×10−43.207067004117469 ×10−43.2070743668963542 ×10−4
15 2.240137639332812 ×10−72.240137681772090 ×10−72.2403482815477269 ×10−7
22 −3.482702552773726 ×10−13 −3.482702562831942 ×10−13 −3.4827117098878988 ×10−13
23 2.457210719924869 ×10−15 2.457208785075626 ×10−15 2.4578986477356111 ×10−15
24 −1.386199675011690 ×10−4−1.386199583437589 ×10−4−1.3862246603200998 ×10−4
25 −1.123708543004768 ×10−7−1.123708508292927 ×10−7−1.1236385421649195 ×10−7
33 −8.404303395022170 ×10−14 −8.404303328191928 ×10−14 −8.4043404516964220 ×10−14
34 1.966253560794704 ×10−41.966253482908411 ×10−41.9662681422586747 ×10−4
35 1.850871822783363 ×10−71.850871866296891 ×10−71.8506905085345953 ×10−7
44 −3.346553299576044 ×106−3.34655331275473 ×106−3.3465596382245759 ×106
45 −7.696119549791547 ×102−7.69611965386665 ×102−7.6960757074656776 ×102
55 −1.323349050039724 ×102−1.32334904512777 ×102−1.3233341260687896 ×102
Table 3: De i a i e o G een’s unc ion UJK,3a e alua ion poin x=(1,1,1)T o MEE
Ma e ial C.
In o de o e alua e he pe o mance o he double Fou ie se ies app oach,
he ollowing e o s schemes a e de ised:
e(K,α)
in (λIJ) :=
α
/
m=−α
α
/
n=−α000λ(m,n)
IJ −λR(m,n)
IJ 000
2α2,(40)
whe e λR(m,n)
IJ is a e e ence ma ix coefficien e alua ed wi h a high numbe
o Gauss abscissas, and λ(m,n)
IJ is he same coefficien being e alua ed wi h a
16
(JK)Fini e diffe ence wi h Bu oni & Sáez (2010) p esen wo k (Fou ie Se ies ρ=2)
11 1.7612993919238762 ×10−41.7613069813436755 ×10−4
12 1.4372187047786456 ×10−41.4372186057878247 ×10−4
13 −2.4864503755635603 ×10−4−2.4864148494725331 ×10−4
14 4.3863676517483414 ×10−44.3861250202513018 ×10−4
15 −5.3289261534288785 ×10−4−5.3288733367096404 ×10−4
22 −3.8721191709814931 ×10−4−3.8719435901746122 ×10−4
23 −2.6527046054561354 ×10−4−2.6527176860501949 ×10−4
24 −6.1505992611293401 ×10−4−6.1506203927697429 ×10−4
25 3.6085826796324048 ×10−43.6084569437355527 ×10−4
33 −5.5435380481296344 ×10−8−5.5444351257591942 ×10−8
34 −1.9217495103716011 ×10−7−1.9217332626512755 ×10−7
35 −2.8879775065048592 ×10−7−2.8879064153501508 ×10−7
44 1.3028359742552897 ×10−71.3028391615667287 ×10−7
45 2.1695644783532599 ×10−82.1693234709859794 ×10−8
55 −1.8092540754710937 ×10−7−1.8092827923235988 ×10−7
Table 4: Second de i a i e o G een’s unc ion UJK,12 a e alua ion poin x=(1,1,1)T
o MEE Ma e ial C.
Ko de ule; and
eS(UIJ) := 1
1UIJ −UR
IJ1
1S
∥UR
IJ∥S
,(41)
whe e ∥·∥S=2S2|·|dω,S2deno es a uni sphe e in R3.
The e o e(K,α)
in de ined in equa ion (40) is used in o de o measu e he
accumula ed e o in he in eg a ion o he Fou ie coefficien s o a gi en
numbe o e ms α.Figu es2-7show hee olu iono hemeane o o all
N×Ncomponen s (N=3,4o 5 o elas ic,PEo MEE)o heFou ie
coefficien s as
'e(K,α)
in :=
N
/
I=1
N
/
J=1
e(K,α)
in (λIJ)
N2.(42)
The e e ence solu ion is compu ed wi h 20 e ms and 256 Gausspoin s
a e used in o de o ob ain Fou ie coefficien s (21). Figu e 2 e e s o PE
Ma e ial A wi h ρ=1.In igu e2(a)weshowinlogscale hemeane o
s. numbe o Gauss poin s used o compu e each o he Fou ie coefficien s
(21). Each cu e co esponds o a αnumbe o e ms included. The same
da a a e shown in igu e 2 (b), his ime as a amily o cu es o cons an
numbe o KGauss poin s as a unc ion o he numbe o e ms α.I can
be obse ed ha i 64 Gauss poin s a e used he e o emains below 10−10
o any numbe o e ms included in he se ies. Figu e 3 shows he same
e olu ion o he e o o he same ma e ial bu using he o mula ion wi h
17
Figu e 2: Mean e(K,α)
in o 16 componen s he undamen al solu ion o Ma e ial A. Fo -
mula ion o ρ=1.(a)E o s.numbe o Gausspoin s.(b)E o s.numbe o e ms
in he se ies
ρ=2.Compa isonwi hp e ious igu e2lead o heconclusion ha less
Gauss poin s a e needed o he same accu acy i he p esen o mula ion
wi h ρ=2 o he Fou ie se ies app oach is used. Figu es 4-7 p esen s
simila esul s o ma e ials B and C. These nume ical es s sugges ha – o
p ac ical applica ions– 64 Gauss poin s a e sufficien o e alua e accu a ely
he Fou ie coefficien s (21), in ag eemen wi h [29].
In o de o show he con e gence beha iou o he Fou ie se ies app oach
o ρ=1and ρ=2, igu es8-10illus a einlogscale hemean alueso
he e o /eS o Ma e ials A, B and C, espec i ely. In his case he e o
is de ined as
'eS:=
N
/
J=1
J
/
I=1
eS(UIJ)
1
2N(N+1) .(43)
Fo all cases, Fou ie coefficien s (21) ha e been compu ed wi h 128 Gauss
poin s. As a e e ence o compu ing eS he non-degene a e solu ion gi en
by (7), (18) and (19) has been used. The in eg a ion on he uni sphe eS2
has been pe o med wi h s anda d double Gaussian quad a u ewi h64×64
poin s. The h ee igu es illus a e he as con e gence ha shows he
Fou ie expansion by aking ρ=2when compa ed wi h con e gence o
ρ=1.I isshown ha highe accu acyisob ained o agi encu -offo
he se ies, o equi alen ly, o a gi en deg ee o accu acy less e ms need o
be included in o he se ies by aking in o accoun his simplede ailon he
pe iodici y o he Ba ne -Lo he enso .
18
Figu e 3: Mean e(K,α)
in o 16 componen s he undamen al solu ion o Ma e ial A. Fo -
mula ion o ρ=2.(a)E o s.numbe o Gausspoin s.(b)E o s.numbe o e ms
in he se ies
Figu e 4: Mean e(K,α)
in o 25 componen s he undamen al solu ion o Ma e ial B. Fo -
mula ion o ρ=1.(a)E o s.numbe o Gausspoin s.(b)E o s.numbe o e ms
in he se ies
19
Figu e 5: Mean e(K,α)
in o 25 componen s he undamen al solu ion o Ma e ial B. Fo -
mula ion o ρ=2.(a)E o s.numbe o Gausspoin s.(b)E o s.numbe o e ms
in he se ies
Figu e 6: Mean e(K,α)
in o 25 componen s he undamen al solu ion o Ma e ial C. Fo -
mula ion o ρ=1.(a)E o s.numbe o Gausspoin s.(b)E o s.numbe o e ms
in he se ies
20

Figu e 7: Mean e(K,α)
in o 25 componen s he undamen al solu ion o Ma e ial C. Fo -
mula ion o ρ=2.(a)E o s.numbe o Gausspoin s.(b)E o s.numbe o e ms
in he se ies
Figu e 8: Mean alues /eSo he 10 diffe en componen s he undamen al solu ion o
Ma e ial A. Fo mula ion o ρ=1,2.
21
Figu e 9: Mean alues /eSo he 15 diffe en componen s o he undamen al solu ion
o Ma e ial B. Fo mula ion o ρ=1,2.
Figu e 10: Mean alues /eSo he 15 diffe en componen s o he undamen al solu ion
o Ma e ial C. Fo mula ion o ρ=1,2.
22
5. Conclusions
The 3D ex ended displacemen undamen al solu ion and i s i s - and
second-o de de i a i es o PE, PM and MEE ma e ials ha e been ob ained
and i s effec i e implemen a ion u he discussed in his pape . The new
exp ession o he undamen al solu ion is (i)explici in e mso heS oh’s
eigen alues, (ii)i emainswell-de inedwhensomeS oh’seigen aluesa e
equal (ma hema ical degene acy) o nea ly equal (quasi-ma hema ical de-
gene acy), and (iii)i isexac . We ealise ha hema hema icaldegene acy
p esen ed in p e ious explici o mula ions can be emo ed by ac o iza ion
o he denomina o in eq. (16), since his comes om he ma hema ical s uc-
u e o he solu ion and no om physical a gumen s. This solu ion is used
as building block o he de elopmen o an al e na i e efficien app oach
based on double Fou ie se ies ep esen a ions. The Fou ie se ies ep esen-
a ion is possible due o he pe iodic na u e o he solu ion.Themainbene i
om his se ies solu ion is ha he in o ma ion o he ma e ial p ope ies is
con ained only in he Fou ie coefficien s, while he in o ma ion o he depen-
dence o he e alua ion poin posi ion is con ained in simple igonome ic
unc ions. This esul s in wo ad an ages: i s , any o de de i a i es can be
de e mined by simple spa ial diffe en ia ion o he igonome ic unc ions.
We p esen esul s o i s - and second-o de bu highe -o de de i a i es can
be buil in a s aigh o wa d way using his me hodology i equi ed. Second,
he Fou ie coefficien s need o be ob ained only once o a gi en ma e ial,
leading o a e y efficien me hodology o nume ical implemen a ions. We
ha e shown ha , exploi ing he π-pe iodici y o a iable φ,be e accu acyis
ob ained o a gi en numbe o e ms, i.e. con e gence is imp o ed (in some
cases by se e al o de s o magni ude). Fou ie expansion ep esen a ion is
eal- alued, which is an impo an ea u e o nume ical applica ions. The
obus ness o he scheme a ise om he ac ha , due o he p ope ies (i)
and (ii)o henewexp ession o hedisplacemen undamen alsolu ion,
he Fou ie coefficien s can be compu ed o any gene al aniso opic coupled
ma e ial wi h any kind o ma hema ical degene acy in he S oh con ex .
In summa y, undamen al solu ions o aniso opic ma e ials deal wi h
wo main d awbacks, he ma hema ical degene acy and i s o e ly complex
and compu a ionally expensi e s uc u e. In his wo k we de eloped a scheme
o he e alua ion o 3D ully aniso opic undamen al solu ion o MEE ma-
e ials me ging he bes o wo wo lds: he clean s uc u e o heS oh o -
malism along wi h he simplici y o Fou ie expansions. These de elopmen s
23
a e expec ed o help o mi iga e he men ioned old d awback o undamen al
solu ions.
Acknowledgemen s
The au ho s would like o dedica e his wo k o he memo y o P o . Ca -
los A. B ebbia (1938-2018), whose ele an esea ch con ibu ions helped o
o ge he Bounda y Elemen Me hod and we e key in i s u he de elopmen .
This wo k was suppo ed by he Minis e io de Economía, Indus ia y
Compe i idad o Spain unde p ojec DPI2017-89162-R. R.J.M. would like
o acknowledge CNPq, p ojec no.310649, o he suppo .
Appendix A. Compu a ion o he S oh’s eigen alues
In his wo k, he S oh’s eigen alues a e ob ained nume ically by sol ing
he linea eigen-p oblem [26]:
)N1N2
N3NT
1*)a
b*=p)a
b*,(A.1)
whe e
N1=−T−1RT,N2=T−1,N3=RT−1RT−Q,(A.2)
wi h Q,Rand Tde ined in (13) and he supe sc ip Tdeno ing anspose.
In his implemen a ion he GEEVX sub ou ine o LAPACK lib a y has been
used in o de o compu e he co esponding eigen alues. Then, he i e
complex eigen alues wi h posi i e imagina y pa a e he so-called S oh’s
eigen alues. The emainde i es a e hei complex conjuga es.
Appendix B. Ma e ials
In his appendix we summa ize he ma e ial p ope ies used in hiswo k.
Ma e ials A and B a e ans e sely iso opic and he e o e sa is ied he ol-
lowing ela ions:
c1212 =c1111 −c1122
2,c
1313 =c2323,c
2222 =c1111,c
2233 =c1133
e322 =e311,e
223 =e113,q
322 =q311,q
223 =q113 (B.1)
λ22 =λ11,ϵ22 =ϵ11,µ
22 =µ11.
Non- anishing componen s o Ma e ials A, B and C a e p esen ed in Tables
B.5, B.6 and B.7, espec i ely.
24