Topological Sensitivity Analysis for Three-dimensional Linear Elasticity Problem

Transcription

1 6 th Worl Congress on Structural an Multscplnary Optmzaton Ro e Janero, 30 May - 03 June 2005, Brazl Topologcal Senstvty Analyss for Three-mensonal Lnear Elastcty Problem A.A. Novotny 1, R.A. Fejóo 1, E. Taroco 1 & C. Para 2 1 Laboratóro Naconal e Computação Centífca LNCC/MCT, Av. Getúlo Vargas 333, Petrópols - RJ, Brasl 2 Centro Atómco Barloche, 8400 Barloche, Argentna 1. Abstract In ths work we use the Topologcal-Shape Senstvty Metho to obtan the topologcal ervatve for three-mensonal lnear elastcty problems, aoptng the total potental energy as the cost functon an the equlbrum equaton as the constrant. Ths metho, base on classcal shape senstvty analyss, leas to a systematc proceure to compute the topologcal ervatve. In partcular, frstly we present the mechancal moel, later we perform the shape ervatve of the corresponng cost functon an, fnally, we compute the fnal expresson for the topologcal ervatve usng classcal asymptotc analyss aroun sphercal cavtes. 2. Keywors: topologcal ervatve, shape ervatve, three-mensonal elastcty, topology esgn. 3. Introucton The topologcal ervatve has been recognze as a promsng tool to solve topology optmzaton problems (see [1, where 425 references concernng topology optmzaton of contnuum structures are nclue. Nevertheless, ths concept s wer. In fact, the topologcal ervatve may also be apple to nverse problems an to smulate physcal phenomena wth changes on the confguraton of the oman of the problem. In aton, extenson of the topologcal ervatve n orer to nclue arbtrary shape holes an ts applcatons to Laplace, Posson, Helmoltz, Naver, Stokes an Naver-Stokes equatons are evelope by Masmou an hs co-workers an by Sokolowsky an hs co-workers. See, for nstance, [10 for applcatons of the topologcal ervatve to the above equatons, nverse problems an materal propertes characterzaton. On the other han, although the topologcal ervatve s extremely general, ths concept may become restrctve ue to mathematcal ffcultes nvolve n ts calculaton. In orer to overcome ths problem, the authors propose an alternatve metho to compute the topologcal ervatve base on classcal shape senstvty analyss. Ths approach, calle Topologcal-Shape Senstvty Metho, was alreay apple for several engneerng problems. Our am n the present paper s to apply the Topologcal-Shape Senstvty Metho to compute the topologcal ervatve n three-mensonal lnear elastcty. Ths ervatve can be apple n several engneerng problems such as structural topology optmzaton. For the sake of completeness, n Secton 4 we present a short escrpton of the Topologcal-Shape Senstvty Metho. In Secton 5 we use ths approach to compute the topologcal ervatve for the problem uner conseraton. In partcular, n Secton 5.1 we present the mechancal moel assocate to three-mensonal lnear elastcty. In Secton 5.2 we compute the shape ervatve for ths problem aoptng the total potental energy as the cost functon an the weak form of the state equaton as the constrant. Fnally, n Secton 5.3, we compute the fnal expresson for the topologcal ervatve usng classcal asymptotc analyss aroun sphercal cavtes. 4. Topologcal-Shape Senstvty Metho Let us conser an open boune oman Ω R 3 wth a smooth bounary Ω. If the oman Ω s perturbe by ntroucng a small hole at an arbtrary pont ˆx Ω, we have a new oman Ω = Ω B, whose bounary s enote by Ω = Ω B, where B = B B s a ball of raus centere at pont ˆx Ω. Therefore, we have the orgnal oman wthout hole Ω an the new one Ω wth a small hole B as shown n fg. (1. Thus, conserng a cost functon ψ efne n both omans, ts topologcal ervatve s gven by ψ (Ω ψ (Ω D T (ˆx = lm, (1 0 f ( 1

2 where f ( s a negatve functon that ecreases monotoncally so that f ( 0 wth 0 +. n x^ n x^ B Fgure 1: topologcal ervatve concept Recently an alternatve proceure to compute the topologcal ervatve, calle Topologcal-Shape Senstvty Metho, have been ntrouce by the authors (see, for nstance, [12. Ths approach makes use of the whole mathematcal framework (an results evelope for shape senstvty analyss (see, for nstance, the poneer work of Murat & Smon [9. The man result obtane n [12 may be brefly summarze n the followng Theorem (see also [11: Theorem 1 Let f ( be a functon chosen n orer to 0 < D T (ˆx <, then the topologcal ervatve gven by eq. (1 can be wrtten as 1 D T (ˆx = lm 0 f ( τ ψ (Ω τ, (2 where τ R + s use to parameterze the oman. That s, for τ small enough, we have Ω τ := { x τ R 3 : x Ω, x τ = x + τv, x τ = x, Ω τ = Ω. (3 In aton, conserng that n s the outwar normal unt vector (see fg. 1, then we can efne the shape change velocty v, whch s a smooth vector fel n Ω assumng the followng values on the bounary Ω { v = n on B (4 v = 0 on Ω an the shape senstvty of the cost functon n relaton to the oman perturbaton characterze by v s gven by τ ψ (Ω τ ψ (Ω τ ψ(ω = lm. (5 τ 0 τ Proof. The reaer ntereste n the proof of ths result may refer to [12 Observe that the topologcal ervatve gven by eq. (1 can be seen as an extenson of classcal shape ervatve, but wth the mathematcal ffculty concernng the lack of homeomorphsm between Ω an Ω. On the other han, the above Theorem hghlghts that the topologcal ervatve may be obtane by means of shape senstvty analyss. Consequently, Topologcal-Shape Senstvty Metho leas to a smple an constructve approach to compute the topologcal ervatve of the cost functon ψ conserng eq. (2. In fact, the omans Ω an Ω τ have the same topology, that allow us to bul an homeomorphc map between them. In aton, Ω an Ω τ can be respectvely seen as the materal an the spatal confguratons. Therefore, n orer to compute the shape ervatve of the cost functon (see eq. 5 we can use classcal results from Contnuum Mechancs lke the Reynols transport theorem an the concept of materal ervatves of spatal fels [5. Fnally, n ths work we wll show these features n the context of three-mensonal elastcty. 2

3 5. The topologcal ervatve n three-mensonal lnear elastcty Now, to hghlght the potentaltes of the Topologcal-Shape Senstvty Metho, t wll be apple to three-mensonal lnear elastcty problems conserng the total potental energy as the cost functon an the equlbrum equaton n ts weak form as the constrant. Therefore, conserng the above problem, frstly we ntrouce the mechancal moel, later we perform the shape senstvty of the aopte cost functon wth respect to the shape change of the hole an fnally we compute the assocate topologcal ervatve Mechancal moel In ths work, we conser a mechancal moel restrcte to smalls eformaton an splacement an for the consttutve relaton we aopt an sotropc lnear elastc materal. Ths assumptons leas to the classcal three-mensonal lnear elastcty theory [4. In orer to compute the topologcal ervatve assocate to ths problem, we nee to state the equlbrum equatons n the orgnal oman Ω (wthout hole an n the new one Ω (wth hole. Problem formulaton n the orgnal oman wthout hole: The mechancal moel assocate to the threemensonal lnear elastcty problem can be state n ts varatonal formulaton as followng: fn the splacement vector fel u U, such that T(u E(η = q η η V, (6 Ω Γ N where Ω represents a eformable boy wth bounary Ω = Γ N Γ D, such that Γ N Γ D =, submtte to a set of surface forces q on the Neumann bounary Γ N an splacement constrants ū on the Drchlet bounary Γ D. Therefore, assumng that q L 2 (Γ N, the amssble functons set U an the amssble varatons space V are gven, respectvely, by U = { u H 1 (Ω : u = u on Γ D, (7 V = { η H 1 (Ω : η = 0 on Γ D. (8 In aton, the lnearze Green eformaton tensor E(u an the Cauchy stress tensor T(u are efne as E(u = 1 ( u + u T := s u an T(u = CE(u = C s u, (9 2 where C = C T s the elastcty tensor, that s, snce I an II respectvely are the secon an forth orer entty tensors, E s the Young s moulus an ν s the Posson s rato, we have C = E (1 + ν (1 2ν [(1 2ν II + ν (I I C 1 [(1 + ν II ν (I I. (10 E The Euler-Lagrange equaton assocate to the above varatonal problem, eq. (6, s gven by the followng bounary value problem: fn u such that v T(u = 0 n Ω. (11 u = ū on Γ D T(un = q on Γ N Problem formulaton n the new oman wth hole: The problem state n the orgnal oman Ω can also be wrtten n the oman Ω wth a hole B. Therefore, assumng null forces on the hole, we have the followng varatonal problem: fn the splacement vector fel u U, such that T (u E (η = q η η V. (12 Ω Γ N 3

4 where the set U an the space V are respectvely efne as U = { u H 1 (Ω : u = u on Γ D, (13 V = { η H 1 (Ω : η = 0 on Γ D. (14 As seen before, the tensors E (u an T (u are respectvely gven as E (u = s u an T (u = C s u, (15 where the elastcty tensor C s efne n eq. (10. In accorance wth the varatonal problem gven by eq. (12, the natural bounary conton on B s T (u n = 0 (homogeneous Neumann conton. Therefore, the Euler-Lagrange equaton assocate to ths new varatonal problem s gven by the followng bounary value problem: fn u such that v T (u = 0 n Ω u = ū on Γ D. (16 T (u n = q on Γ N T (u n = 0 on B 5.2. Shape senstvty analyss Let us choose the total potental energy store n the elastc sol uner analyss as the cost functon. For smplcty, we assume that the external loa remans fxe urng the shape change. As t s wellknown, fferent approaches can be use to obtan the shape ervatve of the cost functon. However, n our partcular case, as the cost functon s assocate wth the potental of the state equaton, the rect fferentaton metho wll be aopte to compute ts shape ervatve. Therefore, conserng the total potental energy alreay wrtten n the confguraton Ω τ, eq. (3, then ψ(ω τ := J τ (u τ can be expresse by J τ (u τ = 1 T τ (u τ E τ (u τ q u τ, (17 2 Ω τ Γ N where the tensors E τ (u τ an T τ (u τ are respectvely gven by wth τ ( use to enote E τ (u τ = s τ u τ an T τ (u τ = C s τ u τ, (18 τ ( := x τ (. (19 In aton, u τ s the soluton of the varatonal problem efne n the confguraton Ω τ, that s: fn the splacement vector fel u τ U τ such that T τ (u τ E τ (η τ = q η τ η τ V τ, (20 Ω τ Γ N where the set U τ an the space V τ are efne as U τ = { u τ H 1 (Ω τ : u τ = u on Γ D, (21 V τ = { η τ H 1 (Ω τ : η τ = 0 on Γ D. (22 Observe that from the well-known termnology of Contnuum Mechancs, the omans Ω τ = Ω an Ω τ can be nterprete as the materal an the spatal confguratons, respectvely. Therefore, n orer to compute the shape ervatve of the cost functon J τ (u τ, at τ = 0, we may use the Reynols transport theorem an the concept of materal ervatves of spatal fels, that s [5 = ϕ τ τ + ϕ τ (v n, (23 Ω Ω Ω τ ϕ τ 4

5 where ϕ τ s a spatal scalar fel an ( s use to enote ( := ( τ = ( τ xτ fxe. (24 Takng nto account the cost functon efne through eq. (17 an assumng that the parameters E, ν, ū, an q are constants n relaton to the perturbaton represente by τ, we have, from eq. (23, that τ J τ (u τ = 1 (T (u E (u (v n Ω 2 τ (T τ (u τ E τ (u τ q u, (25 Γ N where ( s use to enote In aton, u can be wrtten as where ( := ( τ Ω. (26 u = u + ( u v u = u ( u v. (27 Takng nto account the notaton ntrouce through eq. (24 an from eq. (27, we have τ (T τ (u τ E τ (u τ = 2T (u E (u = 2 (T (u E ( u T (u E (ϕ, (28 ϕ = ( u v E (ϕ = s ϕ. (29 Substtutng eq. (28 n eq. (25 we obtan τ J τ(u τ = 1 (T (u E (u (v n T (u E (ϕ 2 Ω Ω + T (u E ( u q u Ω Γ N = 1 (T (u E (u (n v T (u E (ϕ 2, (30 Ω Ω snce u V an u s the soluton of eq. (12. In aton, we observe that T (u E (ϕ = T (u ϕ n v(t (u ϕ. (31 Ω Ω Ω Conserng ths last result (eq. 31 n eq. (30 an takng nto account that u s the soluton of eq. (16, we have τ J τ (u τ = 1 (T (u E (u (v n T (u ϕ 2 n Ω Ω [ 1 = 2 (T (u E (u I ( u T T (u n v, (32 Ω rememberng that ϕ s gven by eq. (29. Let us efne Σ as the Eshelby energy-momentum tensor (see, for nstance, [2, 6 gven n ths partcular case by Σ = 1 2 (T (u E (u I ( u T T (u, (33 then the shape ervatve of the cost functon J τ (u τ efne through eq. (17, at τ = 0, can be wrtten as τ J τ (u τ = Σ n v, (34 Ω 5

6 whch becomes an ntegral efne on the bounary Ω. In other wors, f the velocty fel v s smooth enough n the oman Ω, then the shape senstvty of the problem only epens on the efnton of ths fel on the bounary Ω Topologcal ervatve calculaton In orer to compute the topologcal ervatve usng the Topologcal-Shape Senstvty Metho, we nee to substtute eq. (34 n the result of Theorem 1 (eq. 2. Therefore, from the efnton of the velocty fel (eq. 4 an conserng the shape ervatve of the cost functon (eq. 34, we have that τ J τ (u τ = Σ n n, (35 B where Σ n n = 1 2 T (u E (u T (u n ( u n. (36 In aton, takng nto account homogeneous Neumann bounary conton on the hole, we have, from eq. (16, that T (u n = 0 on B, therefore τ J τ (u τ = 1 T (u E (u. (37 2 B Fnally, substtutng eq. (37 n the result of the Theorem 1 (eq. 2, the topologcal ervatve becomes D T (ˆx = 1 2 lm 1 0 f T (u E (u. (38 ( B Conserng the nverse of the consttutve relaton E (u = C 1 T (u (see eq. ntegran of eq. (38 may be expresse as a functon of the stress tensor as followng T (u E (u = 1 E 10, then the [ (1 + ν T (u T (u ν (trt (u 2. (39 Let us ntrouce a sphercal coornate system (r, θ, ϕ centere n ˆx (see fg. 2, then the stress tensor T (u = (T (u T, when efne on the bounary B, can be ecompose as T (u B = T rr + T rθ + T rϕ (e r e r + T rθ (e r e θ + T rϕ (e r e ϕ (e θ e r + T θθ (e θ e θ + T θϕ (e ϕ e r + T θϕ where e r, e θ an e ϕ are the bass of the sphercal coornate system such that (e θ e ϕ (e ϕ e θ + T ϕϕ (e ϕ e ϕ, (40 e r e r = e θ e θ = e ϕ e ϕ = 1 an e r e θ = e r e ϕ = e θ e ϕ = 0. (41 Snce we have homogeneous Neumann bounary conton on B, then T (u n = 0 T (u e r = 0 on B. (42 From the ecomposton of the stress tensor shown n eq. (40 an takng nto account eqs. (41,42, we observe that T (u e r = T rr e r + T rθ e θ + T rϕ e ϕ = 0 T rr = T rθ = T rϕ = 0. (43 Substtutng eqs. (40,43 nto eq. (39, the topologcal ervatve gven by eq. (38 may be wrtten n terms of the components of the stress tensor n sphercal coornate, as followng D T (ˆx = 1 2E lm 1 0 f T (T θθ (, T θϕ, T ϕϕ B = 1 2E lm 1 2π ( π 0 f T (T θθ (, T θϕ, T ϕϕ 2 sn θθ ϕ, (

7 where T (T θθ, T θϕ, T ϕϕ = (T θθ 2 + (T ϕϕ 2 2νT θθ T ϕϕ + 2(1 + ν(t θϕ 2. (45 Now, t s enough to calculate the lmt 0 n the eq. (44 to obtan the fnal expresson of the topologcal ervatve. Thus, an asymptotc analyss [7 shall be performe n orer to know the behavor of the soluton T θθ, T θϕ an T ϕϕ when 0. Ths behavor may be obtane from the analytcal soluton for a stress strbuton aroun a sphercal vo n a three-mensonal elastc boy [13, whch s gven, for any δ > 0 an at r =, by (see Appenx T θθ T θϕ B = B = T ϕϕ B = 3 { σ1 (u [ 3 5(1 2ν cos 2ϕ + 10 cos 2θ sn 2 ϕ 4(7 5ν +σ 2 (u [ 3 + 5(1 2ν cos 2ϕ + 10 cos 2θ cos 2 ϕ +σ 3 (u [2(4 5ν 10 cos 2θ + O( 1 δ, (46 15(1 ν 2(7 5ν (σ 1 (u σ 2 (u cos θ sn 2ϕ + O( 1 δ, (47 3 { σ1 (u [ 8 5ν + 5(2 ν cos 2ϕ + 10ν cos 2θ sn 2 ϕ 4(7 5ν +σ 2 (u [ 8 5ν 5(2 ν cos 2ϕ + 10ν cos 2θ cos 2 ϕ 2σ 3 (u (1 + 5ν cos 2θ + O( 1 δ, (48 where σ 1 (u, σ 2 (u an σ 3 (u are the prncpal stress values of the tensor T (u, assocate to the orgnal oman wthout hole Ω (see eq. 6, evaluate n the pont ˆx Ω, that s T (u ˆx. Substtutng the asymptotc expanson gven by eqs. (46,47,48 n eq. (44 we observe that functon f ( must be chosen such that f ( = B = 4π 2 f ( = B = 4 3 π3 (49 n orer to take the lmt 0 n eq. (44. Therefore, from ths choce of functon f ( shown n eq. (49, the fnal expresson for the topologcal ervatve becomes a scalar functon that epens on the soluton u assocate to the orgnal oman Ω (wthout hole, that s (see also [3, 8: n terms of the prncpal stress values σ 1 (u, σ 2 (u an σ 3 (u of tensor T (u D T (ˆx = 3(1 ν [ ( 10(1 + ν σ 1 (u 2 + σ 2 (u 2 + σ 3 (u 2 (1 + 5ν (σ 1 (u + σ 2 (u + σ 3 (u 2 ; 4(7 5νE (50 n terms of the stress tensor T (u D T (ˆx = 3(1 ν [ 10(1 + νt (u T (u (1 + 5ν(trT (u 2 ; (51 4(7 5νE n terms of the stress T (u an stran E (u tensors [ 3(1 ν D T (ˆx = 10T (u E (u 1 5ν trt (u tre (u 4(7 5ν 1 2ν, (52 whch was obtane from a smple manpulaton conserng the consttutve relaton gven by eq. (9. See also eq. (10. Remark 2 It s nterestng to observe that f we take ν = 1/5 n eq. (52, the fnal expresson for the topologcal ervatve n terms of T (u an E (u becomes D T (ˆx = T (u E (u. (53 7

8 6. Conclusons In ths work, we have compute the topologcal ervatve n three-mensonal lnear elastcty takng the total potental energy as the cost functon an the state equaton n ts weak form as the constrant. The relatonshp between shape an topologcal ervatves was formally establshe n Theorem 1, leang to the Topologcal-Shape Senstvty Metho. Therefore, results from classcal shape senstvty analyss coul be use to compute the topologcal ervatve n a smple an constructve way. In partcular, we have obtane the explct formula for the topologcal ervatve for the problem uner conseraton gven by eqs. (50,51,52, whose result can be apple n several engneerng problems such as topology optmzaton of three-mensonal lnear elastc structures. Acknowlegments Ths research was partly supporte by CONICET (Argentna an the brazlan agences FAPERJ (E- 26/ /2003 an CNPq/FAPERJ-PRONEX (E-26/ /2003. The support from these agences s greatly apprecate. 7. References [1 H.A. Eschenauer & N. Olhoff. Topology Optmzaton of Contnuum Structures: A Revew. Apple Mechancs Revew, 54: , [2 J.D. Eshelby. The Elastc Energy-Momentum Tensor. Journal of Elastcty, 5: , [3 S. Garreau, Ph. Gullaume & M. Masmou. The Topologcal Asymptotc for PDE Systems: The Elastcty Case. SIAM Journal on Control an Optmzaton, 39: , [4 P. German & P. Muller. Introucton à la Mécanque es Mleux Contnus. Masson, [5 M.E. Gurtn. An Introucton to Contnuum Mechancs. Mathematcs n Scence an Engneerng vol Acaemc Press, [6 M.E. Gurtn. Confguratonal Forces as Basc Concept of Contnuum Physcs. Apple Mathematcal Scences vol Sprnger-Verlag, [7 A.M. Il n. Matchng of Asymptotc Expansons of Solutons of Bounary Value Problems. Translatons of Mathematcal Monographs vol AMS, Provence, [8 T. Lewńsk & J. Sokolowsk. Energy change ue to the appearance of cavtes n elastc sols. Internatonal Journal of Sols an Structures, 40: , [9 F. Murat & J. Smon. Sur le Contrôle par un Domane Géométrque. Thess, Unversté Perre et Mare Cure, Pars VI, França, [10 P. Nettaanmäk et all (es.. ECCOMAS 2004, Mn-symposum on Topologcal Senstvty Analyss: Theory an Applcatons, Jyväskylä, Fnlan, [11 A.A. Novotny. Análse e Sensblae Topológca. Ph. D. Thess, LNCC/MCT, Petrópols, Brasl, 2003 ( novotny/prncpal.htm. [12 A.A. Novotny, R.A. Fejóo, C. Para & E. Taroco. Topologcal Senstvty Analyss. Computer Methos n Apple Mechancs an Engneerng, 192: , [13 M.A. Saowsky & E. Sternberg. Stress Concentraton Aroun a Traxal Ellpsoal Cavty. Journal of Apple Mechancs, , June

9 Appenx: asymptotc analyss In ths appenx we present the analytcal soluton for the stress strbuton aroun a sphercal cavty n a three-mensonal lnear elastc boy. Therefore, let us ntrouce a sphercal coornate system (r, θ, ϕ centere n ˆx, as shown n fg. 2. e r e e r n e 3 e 2 e1 x^ Fgure 2: sphercal coornate system (r, θ, ϕ postone n the center ˆx of the ball B. Then, the stress strbuton aroun the sphercal cavty B s gven, for any δ > 0, by where T rr, T rθ, T rϕ, T θθ, T θϕ for = 1 T rr T rθ T rϕ T θθ T θϕ T ϕϕ [ ( σ σ 1 σ 1 σ ν T rr T rθ T rϕ T θθ T θϕ T ϕϕ = T rr 1 + T rr 2 + T rr = T rθ 1 + T rθ 2 + T rθ = T rϕ 1 + T rϕ 2 + T rϕ = T θθ 1 + T θθ 2 + T θθ = T θϕ 1 + T θϕ 2 + T θϕ = T ϕϕ 1 + T ϕϕ 2 + T ϕϕ 3 + O( 1 δ, 3 + O( 1 δ, 3 + O( 1 δ, 3 + O( 1 δ, 3 + O( 1 δ, 3 + O( 1 δ, an T ϕϕ, for = 1, 2, 3, are wrtten, as: + ( 10(5 ν 3 r 3 5 r 5 [7 5ν + 5(1 + ν 3 r r 5 [7 5ν + 5(1 + ν 3 r r 5 r r 5 (54 sn 2 θ sn 2 ϕ, (55 sn 2θ sn 2 ϕ, (56 sn θ sn 2ϕ, (57 [ + (1 + 10ν (14 3 r r 5 10ν + 25(1 2ν 3 + (28 20ν 10(1 2ν 3 r r 5 σ 1 [7 5ν + 5(1 2ν 3 r r 5 σ 1 [28 20ν + (11 10ν ν r ((1 2ν 3 r 3 5 r 5 r 3 95 r 5 cos 2ϕ cos 2θ sn 2 ϕ, (58 cos θ sn 2ϕ, (59 cos 2θ sn 2 ϕ (28 r ν + 5(1 2ν 3 r r 5 cos 2ϕ, (60 9

10 for = 2 T rr T rθ T rϕ T θθ T θϕ T ϕϕ [ ( σ σ 2 σ 2 σ ν + ( 10(5 ν 3 r 3 5 r 5 [7 5ν + 5(1 + ν 3 r r 5 [7 5ν + 5(1 + ν 3 r r 5 r r 5 sn 2 θ cos 2 ϕ, (61 cos 2 ϕ sn 2θ, (62 sn θ sn 2ϕ, (63 [ + (1 + 10ν (14 3 r r ν + 25(1 2ν 3 + (28 20ν 10(1 2ν 3 r r 5 σ 2 [7 5ν + 5(1 2ν 3 r r 5 σ 2 [28 20ν + (11 10ν ν r ((1 2ν 3 r 3 5 r 5 r 3 95 r 5 cos 2ϕ cos 2θ cos 2 ϕ, (64 cos θ sn 2ϕ, (65 (28 r 5 20ν + 5(1 2ν 3 cos 2θ cos 2 ϕ r r 5 cos 2ϕ, (66 for = 3 T rr 3 = T rθ 3 = T rϕ 3 = 0, T θθ 3 = T θϕ 3 = 0, T ϕϕ 3 = σ 3 [ (38 10ν 3 ( 10(5 ν 3 r r 5 r r 5 σ 3 [ + 10(1 + ν 3 r r 5 σ 3 σ 3 sn 2 θ, (67 cos θ sn θ, (68 (69 [(9 15ν (14 3 r r ν 5(1 2ν 3 r r 5 sn 2 θ, (70 (71 [(9 15ν ((1 3 r r ν 3 r 3 5 r 5 sn 2 θ, (72 where σ 1, σ 2 an σ 3 are the prncpal stress values of the tensor T (u, assocate to the orgnal oman wthout hole Ω, evaluate n the pont ˆx Ω, that s T (u ˆx. In other wors, the tensor T (u was agonalze n the followng way 3 T (u ˆx = σ (e e, (73 where σ s the egen-value assocate to the e egen-vector of the tensor T (u ˆx. =1 Remark 3 It s mportant to menton that the stress strbuton for = 1, 2 was obtane from a rotaton of the stress strbuton for = 3. In aton, the ervaton of ths last result (for = 3 can be foun n [13, for nstance. 10