A fast and non-degenerate scheme for the evaluation of the 3D fundamental solution and its derivatives for fully anisotropic magneto-electro-elastic materials

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