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Kirigami Auxetic Pyramidal Core: Mechanical
Properties and Wave Propagation Analysis in Damped
Lattice
Fabrizio Scarpa, Morvan Ouisse, Manuel Collet, K. Saito
To cite this version:
Fabrizio Scarpa, Morvan Ouisse, Manuel Collet, K. Saito. Kirigami Auxetic Pyramidal Core:Mechanical Properties andWave Propagation Analysis in Damped Lattice. Journal of Vibrationand Acoustics, American Society of Mechanical Engineers, 2013, 135, pp.041001-1 - 041001-11.<10.1115/1.4024433>. <hal-00993364>
HAL Id: hal-00993364
https://hal.archives-ouvertes.fr/hal-00993364
Submitted on 20 May 2014
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r♠ ①t ♣②r♠ ♦r ♠♥ ♣r♦♣rts ♥
♣r♦♣t♦♥ ♥②ss ♥ ♠♣ tt
r③♦ r♣∗
♥ ♦♠♣♦sts ♥tr ♦r ♥♥♦t♦♥ ♥ ♥
❯♥rst② ♦ rst♦ rst♦ ❯
♦r♥ ss ♥ ♦t†
♥sttt é♣rt♠♥t é♥q ♣♣qé
❯ s♥ç♦♥ r♥
③② t♦‡
♥sttt ♦ ♥str ♥ ❯♥rst② ♦ ♦②♦ ♣♥
strt
♦r srs t ♠♥tr♥ ♠♥ ♣r♦♣rts ♥ ♣r♦♣t♦♥ r
trsts ♦ ♣②r♠ tt ♠ ①t♥ ♥ ①t ♥t P♦ss♦♥s rt♦ ♦r
♦♥trr② t♦ s♠r tt tsst♦♥s ♣r♦ s♥ ♠t ♦rs t ♣②r♠ tt sr
♥ ts ♦r s ♠♥tr s♥ r♠ r♠ ♣s tt♥ ♣ttr♥ t♥q ♥
♣♣ t♦ r rt② ♦ tr♠♦st ♥ tr♠♦♣st ♦♠♣♦sts t♦ t ♣rtr
♦♠tr② rt tr♦ ts ♠♥tr♥ t♥q t r♠ ♣②r♠ tt s♦ ♥
♥rs♦♥ t♥ ♥♣♥ ♥ ♦t♦♣♥ ♠♥ ♣r♦♣rts ♦♠♣r t♦ ss ♦♥②♦♠
♦♥rt♦♥s ♦♥ ♥t ♣♣r♦①♠t♦♥s r s t♦ t t s♦♥ss rs s♦♥
♥s ③r♦rtr ♣♦♥♦♥ ♣r♦♣rts ♥ t tr♥srs ♣♥ ♥♦ ♣r♦♣t♦♥
t♥q s ♦♥ ♦ s ♦r ♠♣ strtrs s s♦ ♣♣ t♦ t t s♣rs♦♥
♦r ♦ ♦♠♣♦st r♣♦①② tts t ♥tr♥s ②strt ♦ss ♣r♦♣
t♦♥ ♥②ss s♦s ♥s♥ rtt② t r♥t rq♥② ♥ts ♥ ♦♠♣① ♠♦
♦r t♦ ♥s ♦r♠t♦♥ ♠♥s♠ ♦ t tt
∗tr♦♥ rss sr♣rst♦†tr♦♥ rss ♠♦r♥♦ss♥♦♠tr‡tr♦♥ rss st♦st♦②♦♣
❯
tt ♣②r♠ ♦rs ♥ ♦♣ ①t♥s② r♥ t st ②rs ②
rt ♦ ♥♦ ♠t st♥ t♥qs tt ♦ ♣r♦♥ t rt② ♦ ♦st r
♥trt s♥ ♣♥s ❬ ❪ ♥ ♦ t ♠♦r trs ♦ ts tt ♦rs s t
s♣ str♥t t♦ r ♥ ♥♣♥ st♥ss ♥ s t♦ ♦♣ ♥♦
♥r② s♦r♥ ♥trt r strtrs ♦r stt ♥ ②♥♠ ♦♥ ♥ tr♠
♠♥♠♥t ❬ ❪
♣②r♠ tt ♦♥rt♦♥ ♥ tsst ♥ r ②♦t r
s ♣rtr tsst♦♥ s ①t ♥ t ♣♥ ①ts ♥ts r ss ♦ s♦s
♥ strtrs ①t♥ ♥t P♦ss♦♥s rt♦ t s♦ ①♣♥s rtr t♥ ♦♥trts
♥ ♣ ♦♥ ♦♥ rt♦♥ ①ts ❬❪ ♥ ①t♥s② st ♦r tr ♠
t♥t♦♥ ♣r♦r♠♥ ♥ ♣rtr ♦r tr t♥t② ♥ ♥♣ ♦r ❬❪ ♥
r♦st t♦r♥ ♥ ♠♦r♣♥ strtrs ❬ ❪ r tsst♦♥ s ♥ ♥
t ② r♠ t s ♣♦t♥t ♦r♠t♦♥ ♠♥s♠ ♣r♦♥ ①tt② ♥ ♣♦②♠r
♥t♦rs t r♦♠ ①❬❪r♥ ♦s ❬❪ s♠ tsst♦♥ s ♥ ♥t s
①st♥ ♥ ②♣♣rs t s♥ ♥ ♠t r♦♥ ♥♥♦ts ①t♥ s♥
♥rs♦♥ r♦♠ ♣♦st t♦ ♥t P♦ss♦♥s rt♦ ♥ st t♦ ♠♥ ♦♥
❬ ❪
r♠♥s♦♥ ♦♠♣① r s♦s ♥ s♦ ♣r♦ s♥ ♦tr ♠trs
♥ ♣rtr tr♠♦st ♥ tr♠♦♣st ♦♠♣♦sts ② ♠♥s ♦ r♠ t♥qs
r♠ s ♥ ♥♥t ♣♥s rt ♦ ♦♠♥♥ ♣②♦♥ t♥qs r♠ t t
t♥ ♣ttr♥s t♦ ♦t♥ ♥r tsst ♥ ♦♠♣① strtrs r♠ s ♥ s
sss② t♦ ♣r♦ ♦♥②♦♠s t ♥r s♣s s♥ r♦♥ ♥ r ♦♥
rs ❬❪ ①t r♥tr♥t strtrs ❬❪ ♥ ♥♦ ♠♦r♣♥ ♥♦① ♦♥rt♦♥s
❬❪ r s♦s ♦ r♠ rs♦♥ ♦ t ♣②r♠ tt ♥ rt s♥
st♥r r ♣♦①② ♣r♣r♣s ① ♦♠♣♦sts t ❯ sts t
t ♦ ♥ r♠ ♣ttr♥ r t r♦♠ s♥ ♣② ♦ t ♣r♣r ② ♥ t♦♠t ♣②
ttr r sts t ♣♦②t②♥ rs ♠ r ♦ ♥ ♦r s♣s ②
s♥ sqr r♦s ① ① ❬♠♠❪ ♠♥♠ s r r♦s r ♣t
♦♥ t r♦ss ♣♦♥ts ♦ r r♦♠ t ♣♣r s ♥ ♦tr r♦s r ♣ ♦♥
r ♦♥s r♦♠ t s ♦ r tr ♣♥ t♦tr ts r♦s ♥ r
t st s ① ♥ t s♣ ♦ t ①t ♣②r♠ tt ss♠② s ②
t♣s ♥ ♥srt ♥t♦ ♠ rs rr♥♠♥t s ♣t ♥ ♥
t♦ ♦r ♠♥ts t Co t♦ r t ♣r♣r ♥ r s♠♣ s s♦♥ ♥
r
♦r sr ♥ ts ♣♣r s ♦♥r♥ t t t♦♥ ♦ t ♦♥ ♥t
♣r♦♣rts rst ♦rr ♦♠♦♥st♦♥ ♥ t t♦♠♥s♦♥ ♣r♦♣t♦♥ ♦ t
r♠ ♣②r♠ tt ♦♥ ♥t ♣♣r♦①♠t♦♥ s s t♦ r t t
strssstr♥ t♥s♦r ♦ t ♦♠♦♥s tt ♥ t♦ t t s♦♥ss rs r♦♠ t
♥ ♣r♦♠ ss♦t t♦ t rst♦s qt♦♥ ♥ r♦s ♣♥s rt♦♥
♥ t st② ♦ t s♦♥ss rs s t ♥tt♦♥ ♦ ♣♦ss ♥♦♠♦s ♦r
♦ tr rtr rtr ♦ t s♦♥ss sr s ♥ st s ♠trs t♦
♥t② sts ♥ ♣♦♥♦♥ ♠♥ r t sts r ♠sr s ♣r♦ ♥s ♦♥
t s♦♥ss sr ❬❪ ❩r♦s♦♥ss rs r tr♦r s②♠♣t♦♠ ♦ ♣♦ss s♦t♦♥s
rt t t♦r s ♦r♥t ♦♥ ♥s ♦rrs♣♦♥♥ t♦ ts ♥stt②
♣♦♥ts ♦ t st ♦ t t♦rs ♥♦ ♥ ①t♥s st② ♦ t ♦♠♦♥s
♠♥ ♣r♦♣rts rst ♦rr ♦♠♦♥st♦♥ ♥ s♦t♦♥s ♦ t rst♦ qt♦♥
♦ t ♣rtr ♣②r♠ tsst♦♥ s♦♥ ♥ r s ♥♦t ♥ ♣r♦r♠ ②t ❲
s♦ tt t tt s ①t ♥ t ♣♥ t s♦ ♣♦ssss♥ ♣r ♠♥
♦r r t ♥♣♥ st♥ss s r t♥ t tr♥srs ♦♥ ♦♥trr② t♦ ss
♥trs②♠♠tr ♦♥②♦♠ ♦♥rt♦♥s ①st♥ ♥ ♦♣♥ trtr r♠ tt
♦s s♦ ♥ ③r♦ rtr s♦♥ss rs ♥ t tr♥srs ♣♥ ♣♥♥ ♦♥ t
♦♠tr ♣r♠trs ♥♥ t tt ♦r♦r ♦♥sr s♦ t s s ♥ t
r ♥♥r♥ ♣♣t♦♥ r ②strt ♠♣♥ s ♣rs♥t ♥ t s②st♠ t♦
t ♥tr♥s ♦ss t♦r ♦ t ♣r♣r s t♦ ♠♥tr ts tts ♥♦
♣r♦♣t♦♥ t♥q s ♣rs♥t ♥ ts ♦r t♦ ♦♥sr t t ♦ t ♠♣♥ ♥ t
tt ♥ ♥t② t rtt② ♦ ♥s♥ ♠♦s ♣rs♥t t r♥t rq♥s
❲ t ♦♥ ♥t ♣♣r♦①♠t♦♥ ♦s ♥♦t r ♥② ♥s ♣r♦♣t♦♥
♣ttr♥ ♥ t ♣♥ ♦ t tt t ♠♣ ♣r♦♣t♦♥ t♥qs t
t ♦♠♣① ♠♦ ♦r ①st♥ t♥ t ♠♥ts ♦ t ♥t strtr ♥
t ♥s♥ rtt② t♦ s♣ ♠♦s ss♣t♥ ♠♦r ♥r② t♥ t ♦trs
r Pr♦t♦♥ ♦ ♣②r♠ tt ♦r s♥ ♣r♦rt♦♥r♦♥♦♥ t♥q
♦♥ ♠ts r♦♠ ❬❪
r t♣s ♦ t ♠♥tr♥ ♦r t r♠ ①t ♣②r♠ tt ♦r
♥ ♦♠♦♥st♦♥
q♥t ♠♥ ♣r♦♣rts ♦ t r♠ tt strtr ♥ ♦♠
♣t s♥ ♥t♠♥t s ♦♠♦♥st♦♥ t♥qs ♦r ♣r♦ ♠ ❬❪ ♥
♣r♦ ♠♠ t ♥rs ♥♦ s♣♠♥t t♦r u ♥ ①♣rss s
u = ǫx+ u′ r ǫ s ♥ ♣♣ str♥ ♥ u′ s♦② tt♥ ♣r♦ ♥t♦♥
♦ u rt♦♥ tt♦♥ t♦ ♥t② u′ s ①♣rss ② ❬❪
v′T [K] u′ = v′T F
❲r v′ s ♥♦tr ♣r♦ tt♥ ♥t♦♥ s♦♥ qt♦♥ [K] s t
st♥ss ♠tr① ♦ t s②st♠ ♥ F s t♦r ♦ ♥rs ♥♦ ♦rs ♦r♥ t♦
st♥r t♥qs t st♥ss ♠tr① s ♥ s
[K] =∑
e
[k]e =1
V
ˆ
e
[B]T [C] [B] dV
♥ [B] ♦♥t♥s t s♣t rts ♦ t s♣ ♥t♦♥s [C] t s♦♥ ♦rr
strssstr♥ t♥s♦r ♦ t ♠tr ss♦t t♦ t eth ♠♥t ♥ V t ♦♠ ♦ t
♦r ♠♦ ❲♥ ♣rsr str♥ ǫ s ♠♣♦s t ♥rs ♦r t♦r F♥ ①♣rss s
F = −[
K]
ǫ
❲r
[
K]
=∑
e
[
k]
e=
1
V
ˆ
e
[B]T [C] dV
♥srt♥ ♥ ♥ ♦♥ ♦t♥s t ♦♠♦♥s strssstr♥ t♥s♦r rt♦♥s♣
r♣rs♥t♥ t q♥t ♠tr ♥♦s ♥ t ♦♠ V
[
C]
ǫ = σ
❲r σ s t ♦♠ r t♦r ♦ t strsss σij, i, j = 1 . . . 6 ♦r t r♣
rs♥tt ♥t ♦ t ♣r♦ ♠r♦strtr ♦♠♦♥s ♦♠♣♥ ♠tr①
[
S]
=[
C]−1
s q ♦r tr♥srs s♦tr♦♣ ♠tr s
[
S]
=
1Ex
−νyxEx
−νzxEx
0 0 0
−νxyEy
1Ey
−νzyEy
0 0 0
−νxzEz
−νyzEz
− 1Ez
0 0 0
0 0 0 − 1Gxz
0 0
0 0 0 0 1Gyz
0
0 0 0 0 0 1Gxy
r♣rs♥tt ♥t ♠♥t ❱ s ♦r t r♠ ♣②r♠ tt s
s♦♥ ♥ r ♣②r♠ tt s sr s♥ t ♥♦♥♠♥s♦♥ ♣r♠
trs α = a/b β = t/l δ = l/b ♥ ♥tr♥ ♥ θ ♥ s♦tr♦♣ ♥♣♥ ♠♥
♦♥rt♦♥ s ♦r α = 1 ♣r♠tr δ ♥ts ♦ r t ♦r ♦ t
♠♥t l ♣rts r♦♠ ♠ strtr ❲t♥ t ♦♥t①t ♦ ts ♦r ♦♥② ♥
♣♥ s♦tr♦♣ ♦♥rt♦♥s α = 1 ♥ ♦♥sr ♠♦ ♦ t ♥t
s ♥ ♦♣ s♥ t ♦♠♠r ♦ ❨ r r♠♥s♦♥
strtr ①r ♠♥ts t r♥♥ ♥tr♣♦t♦♥ ♥t♦♥s ♥♦s
♥ s ♥ s t♦ r♣rs♥t t ♣②r♠ strtr t ♥♦r♠ ♠s s③
q♥t t♦ t/2 ss♠ tr ♠s ♦♥r♥ tst ♣r♦ ♦♥r② ♦♥t♦♥s
❬❪ ♥ ♣♣ t♦ t s xy ♦♥ t ♥t t ♠♥♠♠ ♥ ♠①♠♠ z
♦♦r♥ts ♥ ♣t r ♦ ♠♣♦s s♣♠♥ts ♦rrs♣♦♥♥ t♦ ♥♦r♠ str♥s
s t♦ r♣rs♥t ♥ ♥♥t ♣r♦ ♦r ♠tr ♦r s♥ strtrs ♥ tr♦r
♣r♦t② ♥ t xy♣♥ ♦♥② ♥t str♥s t t ♦♥r②
ǫx ǫy ǫz γxz γyz γxy
T
♥ ♠♣♦s s s♣♠♥t s ♥ t♦ stt ♥r ♣r♦♠ s♦ t
t♦♥♣s♦♥ s♦r ♣♥♥ ♦♥ t s♣ ♦♠♥t♦♥ ♦ ♥♦♥♠♥s♦♥ ♦♠
tr② ♣r♠trs s t ♠♦s s③s r♥♥ t♥ t♦ s s
♠♥ ♣r♦♣rts ♦ t ♦r r r♣rs♥tt ♦ r ♦♥ r ❬❪
t ❨♦♥s ♠♦s Ec = 30GPa P♦ss♦♥s rt♦ νc = 0.4 ♥ ♥st② ρc = 1600 kgm−3
♦r t s♠t♦♥s t ♥t ♦ t rs l s ss♠ ♦♥st♥t t ♠♠ t
rt ♥st② β s ♠♣♦s q t♦ t♦ ♦ sr ♦r♠t♦♥ ♦♥trt♦♥s r♦♠
t r♦ssst♦♥ ♦ t rs ♥ ss♠ t♥ ♠ ♦r ♣t strtrs ♥ ♣rs♥t
♥ t r♠ tt ♦♥sst♥t t♦ s♠♣s ♣r♦ t t ♠♥tr♥ ♣r♦ss
s♦♥ ♥ r
rst♦s qt♦♥s
s♦♥ss rs ♥ s♠t r♦♠ t s♦t♦♥ ♦ rst♦s qt♦♥s ♦r
♥ k ♣r♦♣t♦♥ rt♦♥ t qt♦♥s t ♦rrs♣♦♥♥ ♥ ♣r♦♠
❬ ❪
[Γ] A = γ A
❲r Γ = kicijklkl ♥ γ = ρv2 cijkl ♥ts t ♦♠♦♥s ♦rt♦rr strss
str♥ t♥s♦r ♦ t q♥t ♠tr ♦ t ♣②r♠ ①t tt tr♠♥♥t
♦ qt t♦ ③r♦ ②s tr ♥s ♥ tr♠s ♦ ρv2 (k) ♥ t♦ tr s♦♥ss
rs s (k) = 1/v (k) ss♦t t♦ |k| = 1 ❬❪ ♦♥sr♥ t t♦r k ♣r♦♣t♥
♥ t xz ♣♥ t k = cos (φ)k+sin (φ) i i,k ♥ t ♥t rs♦rs ♦ t x ♥ z ①s
rs♣t② t ♠tr① Γ♥ ♥ rst ♦r t tr♥srs s♦tr♦♣ ♠r♦strtr s
❬❪
[Γ] =
c55 + (c11 − c55) sin2 φ 0 (c13 + c55) cosφ sinφ
0 c44 + (c66 − c44) sin2 φ 0
(c13 + c55) cosφ sinφ 0 c33 + (c55 − c33) sin2 φ
❲♥ t st ♦♥st♥ts r ♥♦r♠s ♥st c11/√ρ ρ ♥ t ♥st② ♦ t
♦♠♦♥s ♠tr t tr ♥s γi t s♦♥ss rs si (φ) = v−1i (φ) =
γ−1/2i (φ) ♥ t xz ♣♥
♦r ♣r♦♣t♦♥ ♥ t xy ♣♥ k = cos (φ) j+ sin (φ) i t i, j t ♥t
rs♦rs ♦♥ t x ♥ y rt♦♥s rs♣t② t Γ ♠tr① ss♠s t ♦♦♥ ♦r♠
❬❪
[Γ] =
c11 cos2 φ+ c66 sin
2 φ (c12 + c66) cosφ sinφ 0
(c12 + c66) cosφ sinφ c66 cos2 φ+ c22 sin
2 φ 0
0 0 c55 cos2 φ+ c44 sin
2 φ
s♦♥ss rs si (φ) = γ−1/2i (φ) ♥ s♦ t ♥♦r♠s♥ t ♦♥ts
♥ ② c11/√ρ ♥s ♦ ♥ t s♥ t tt r♦t♥
t t t♥s♦r ♦♥ts cij ♦t♥ r♦♠ t ♦♠♦♥st♦♥ ♣r♦r sr
♦
♦qt ♥ ♦ t♦r♠s ♦r st♦②♥♠ s♣rs♦♥ ♥②ss
t♠t r♠♦r
t s ♦♥sr ♥ ♥♥t ♣r♦ st♦②♥♠ ♣r♦♠ s ♣rs♥t ♥ r
r♠♦♥ ♦♠♦♥♦s ②♥♠ qr♠ ♦ s②st♠ s r♥ ② t ♦♦♥ ♣rt
rt qt♦♥
ρ(x)ω2w(x) +∇C(x)∇sym(w(x)) = 0 ∀x ∈ R3
r w(x) ♥ R3 s t s♣♠♥t t♦r C(x) st♥s ♦r t ♦♦ stt② t♥s♦r
♥ ε(x) = ∇sym(w(x)) = 12(∇wT (x) + w(x)∇T ) s t str♥ t♥s♦r ② ♦♥sr♥
♣r♠t ♦ t ♣r♦ ♣r♦♠ ΩR ♥ ② s♥ t ♦ t♦r♠ t ss♦t
♦ ♥♠♦s ♥ t s♣rs♦♥ ♥t♦♥s ♥ ♦♥ ② sr♥ t ♥ s♦t♦♥s
♦ t ♦♠♦♥♦s ♣r♦♠
w(x) = wn,k(x,k)eik.x
r wn,k(x,k) r ΩR♣r♦ ♥t♦♥s ♥ tt s wn,k(x,k) ♥ ωn(k) r t
s♦t♦♥s ♦ t ♦♦♥ ♥r③ ♥s ♣r♦♠ ♦ t st ♦♣rt♦r s
sr ♥
ρ(x)ωn(k)2wn,k(x) +∇C(x)∇sym(wn,k(x))
−iC(x)∇sym(wn,k(x)).k − i∇C(x)12(wn,k(x).k
T + k.wTn,k(x))
+C(x)12(wn,k(x).k
T + k.wTn,k(x)).k = 0 ∀x ∈ ΩR
wn,k(x−R.n)−wn,k(x) = 0 ∀x ∈ ΓR
t s ♦♥sr t ♣rt rt qt♦♥s ♦♥ ♥t ΩR t ①♣rss♦♥
st♥s ♦r ♥r③ ♥ ♣r♦♠ ♥ t♦ ♦♠♣tt♦♥ ♦ t s♣rs♦♥ rs
ωn(k) ♥ ♦rrs♣♦♥♥ ♦qt ♥t♦rs wn,k(x) ♦r ♣♣t♦♥s t
t♦rs r s♣♣♦s t♦ ♦♠♣① ♠♣♥ tr♠s r ♥t♦ qt♦♥ ♥ ♥
tr♦r rtt♥ s k = k
sin(θ) cos(φ)
sin(θ) sin(φ)
cos(θ)
= kΦ r θ ♥ φ r♣rs♥t t rt♦♥
♥s ♥t♦ t r♣r♦ tt ♦♠♥ ♥ Φ s t rt♦♥ t♦r s ♦♠♣♦st♦♥
ss♠s tt r ♥ ♠♥r② ♣rts ♦ t t♦r k r ♦♥r
tr t♥ ♥t♦ ♦♥srt♦♥ t ♣r♦t② ♦♥t♦♥s ♥ rr♥♠♥ts
♦r♠t♦♥ ♦ t ♣r♦♠ ♥ rtt♥ s
∀wn,k(x) ∈ H1(ΩR,C3)/wn,k(x−Rn) = wn,k(x) ∀x ∈ ΓR ,
´
ΩRρ(x)ω2
n(k)wn,k(x)wn,k(x)− εn,k(x)C(x)εn,k(x)
+ikκn,k(x)C(x), εn,k(x)− ikεn,k(x)C(x)κn,k(x)
+k2κn,k(x)C(x)κn,k(x)dΩ = 0
♥♠r ♠♣♠♥tt♦♥ s ♦t♥ ② s♥ st♥r ♥t ♠♥ts ♠t♦ t♦
srts t ♦r♠t♦♥ ss♠ ♠tr① qt♦♥ s ♥ ②
(K + λL(Φ)− λ2H(Φ)− ω2n(λ, (Φ)M )wn,k(Φ) = 0
r λ = ik M ♥ K r t st♥r s②♠♠tr ♥t ♠ss ♥ s②♠♠tr s♠
♥t st♥ss ♠trs rs♣t② L s ss②♠♠tr ♠tr① ♥ H s s②♠♠tr
s♠♥t ♣♦st ♠tr①
M →ˆ
ΩR
ρ(x)ω2n(k)wn,k(x)wn,k(x)dΩ,
K →ˆ
ΩR
εn,k(x)C(x)εn,k(x)dΩ,
L →ˆ
ΩR
−κn,k(x)C(x)εn,k(x) + εn,k(x)C(x)κn,k(x)dΩ,
H →ˆ
ΩR
κn,k(x)C(x)κn,k(x)dΩ
❲♥ k ♥ Φ r ① t s②st♠ s ♥r ♥ ♣r♦♠ t s♦t♦♥ ♦
♦s ♦♠♣t♥ t s♣rs♦♥ ♥t♦♥s ω2n(k,Φ) ♥ t ss♦t ♦ ♥
t♦r wn,k(Φ)
s ♣♣r♦ s ss② s ♦r ♦♠♦♥③t♦♥ s♣tr s②♠♣t♦t ♥②ss ❬❪
♥ t ♦♥t①t ♦ ♣r♦♣t♦♥ s♣rs♦♥ rtrsts ♥ r ♥ ♦qt
♣r♦♣t♦rs r ♣rrr ♦r ♦r qs ♦♠♣tt♦♥ s ♥t ② ❬❪ ❬❪ ♦r
❬❪ ♦st ♦ t t♠ ts t♥qs r ♣♣ ♦r ♥♠♣ ♠♥ s②st♠s
r♣rs♥t ② st ♦ ♦♥st♥t ♥ r ♠trs ss ♣♣r♦ ♦♥ssts ♥ s♥
♠s ♦ t r ks♣ k ♦r λ ♥ Φ ♥s t rst r♦♥ ③♦♥ ♦r ♦t♥♥
t ♦rrs♣♦♥♥ rq♥② s♣rs♦♥ r♠s ♥ t ss♦t ♦qt t♦rs ♦r
♥♠♣ s②st♠s ♦♥② ♣r♦♣t♥ ♦r ♥s♥t s ①st ♦rrs♣♦♥♥ t♦ ♠s ♦
♥s♦t♦♥s ♣r② r ♦r ♠♥r② sr♠♥t♦♥ t♥ ss ♦ s s qt
strt♦rr ♠♣ s②st♠ s ♦♥sr ♠trs K, L, H r ♦♠♣①
t ♥s♥t ♣rt ♦ t ♣r♦♣t♥ s ♣♣r s t ♠♥r② ♣rt ♦ ω2n(λ,Φ) ♥
rs t t♥ ♦♠s r② t t♦ st♥s t t♦ ♠s ♦ s t s♦ t♦
♦♠♣t t ♦rrs♣♦♥♥ ♣②s s ♠♦♠♥ts ② ♣♣②♥ s♣t ♦♥♦t♦♥
t s ♣♦ss t♦ rrr♥ t s②st♠ ♥ ♦rr t♦ t ♥t♦ ♦♥t ♠♣♥ ts ❬❪
s ♦r♠t♦♥ s♦ ♣r♦s st ♦♥t♥ts ♦r t♠s♣ ♦♥♦t♦♥ ♥ ♦r t
♦♠♣tt♦♥ ♦ s♦♥ ♣r♦♣rts s ♥ ② ❬❪ ♦r ❬❪ t ♦♥ssts ♥ ♦♥sr♥ t
♦♦♥ ♥rs ♥ ♣r♦♠
(K − ω2M ) + λn(ω,Φ)L(Φ)− λ2n(ω,Φ)H(Φ))wn,k(Φ) = 0
♥ ts ♣r♦♠ t ♣st♦♥ ω ♥ t ♣r♦♣t ♥ Φ r ① r ♣r♠trs
❲s ♥♠rs λn = ikn ♥ ss♦t ♦qt t♦rs wn,k r t♥ ♦♠♣t ② s♦♥
t qrt ♥ ♣r♦♠ s ♣♣r♦ ♦s t♦ ♥tr♦ rq♥②♣♥♥t
♠trs ♦rrs♣♦♥♥ t♦ ♥r③ ♠♣♥ tr♠s s♦stt② ♠t♣②ss ♦♣♥
tr♦♠♥ t tr♦♥ ♦r♥r② r♥t qt♦♥ ♦♠ ♦tr ♠♦
♦r ♦♣♥ ♦♠♥ ♦♥r② ♦♥t♦♥s ♦♠♠r ♦♥t♦♥ s ♦♥ ts ♣♣r♦ ♥
♥rs ♦rr tr♥s♦r♠t♦♥ ♥ t ks♣ ♦♠♥ s t♦ t t ♣②s s
s♣♠♥ts ♥ ♥r② s♦♥ ♦♣rt♦r ♥ t ♣r♦ strt♦♥ s ♦♥♥t t♦
♥♦tr s②st♠ ❬❪ ♥♦tr t♠♣♦r ♥rs ♦rr tr♥s♦r♠t♦♥ ♥ ♣r♦ ②
t♦ ss s♣t♠♣♦r rs♣♦♥s ♦r ♥♦♥♦♠♦♥♦s ♥t ♦♥t♦♥s s L s s
s②♠♠tr t rst♥ ♥s r qr♣ λ, λ,−λ,−λ ♥ ♦♣s♥ ♥t♦ r
♦r ♠♥r② ♣rs ♦r s♥ ③r♦ ♥ ♠trs r r ♦r ♥♠♣ s②st♠s ♥
t ♥r ♠♣ s ♦♥ ♦t♥s ♣r ♦ r ♥s ♦rrs♣♦♥♥ t♦ ♥s♥t
♠♦s ♦r♥t ♥ t♦ ♦♣♣♦st rt♦♥s ♦♥ t ks♣ ♥ ♠♥r② s t♦ t♦
tr♥ s ♣r♦♣t♥ ♥ ♦♣♣♦st rt♦♥ ♥s♦t♦♥s r s♠r t♦ t♦s
♥ ② t ♠t♦ ♥ ♦r ♦♠♦♥♦s ♠tr ♥♦♥ ♣r♦ ♥ t♦♥
♠♣♦rt♥t ♣r♦♣rts ♥ ①tr♣♦t r♦♠ ❬❪ t② ♦ t ♠t♦ ♣r♦♣♦s
♥ ts ♦r s ♥ ♣r♦ t sr tst ss ♥ ❬❪
s ♣r♦s② ♠♥t♦♥ t r ♣rt ♦ k = kΦ t♦r s ss② rstrt t♦ t
rst r♦♥ ③♦♥ ♥ t qrt ♥ ♣r♦♠ t t② ♦ t ♦♠♣t
s♦t♦♥ s ♥♦ ♦♥r ♦♥♥ t♦ s♣ r♦♥ ③♦♥ ♦r rt♦♥ t♦rΦ ♦rt♦♦♥
t♦ t tt ts ♦r Φp1 = [1, 0]T ♥ Φp2 = [0, 1]T ♥ rt♥r t
♣r♦ ♦♥t♦♥s ①♣rss ♦r ♦♥ ♠♥s♦♥ r st λj(Φp) s
♥ ♥ ss♦t t♦ wj,k(Φp) t♥ ∀m ∈ Z3, λ + i.ΦTp (G.m) s s♦ ♥ ♥
ss♦t t♦ wj,k(Φp)e−i.ΦT
p (G.m)x s ♦r ♥♠♣ s②st♠s t ♥s r
♣r♦② strt ♥ t ks♣ ♦♥ ts ♣r♥♣ rt♦♥s
♦♠♣tt♦♥ ♦ ♥s♥ ♥ ♠♣ ♣♦r ♦ rtr
♥ ♠ ♦ ts ♦r s t♦ ♣r♦ ♥♠r ♠t♦♦♦② t♦ sr t ♣rtr
♦r ♦ t ♥r② ♦ ♥t♦ t ♣r♦② r♠ ①t ♦r ♥ tr♦r ♥s
t♦ ♥ st ♥t♦r ♦r st♥s♥ ♣r♦♣t ♥ ♥s♥t ♦r s♣②
♥ ♠♣ s②st♠ s ♦♥r♥ ♣t② ♦ ♥ ♦ t♦ tr♥s♣♦rt ♥r②
s ♥ ② ts r♦♣ ♦t② ♥ts ♦ ♥r② s tr♥s♣♦rt ♥t♦ t ♦♥sr
s②st♠ ♦♥ s♦ t♦ st♥s t♥ ♣r♦♣t ♥ ♥s♥t s ♦
♥ s♦t♦♥ un(ω, φ), kn(ω) s ♦♥sr t ss♦t r♦♣ ♦t② t♦r ❬❪
s ♥ ②
Cgn(ω, φ) = ∇kω =〈〈S〉〉〈〈etot〉〉
=〈I〉
〈Etot〉
❲r 〈〈:〉〉 s t s♣t ♥ t♠ r rs♣t② ♦♥ ♦♥ ♥ ♦♥ ♣r♦ ♦
t♠ s t ♥st② ♦ ♥r② ♦ I t ♠♥ ♥t♥st② ♥ etot Etot t t♦t ♥r② ♥
ts t♠ r ♦♥ ♣r♦ s ❭t④②s♥♦r⑥ ♦r ts
♥t♥st② t♦r I s ①♣rss s
〈In〉 = −ω
2Re
(ˆ
Ωx
C(εn(x) + ikΞn(x)).(w∗n(x))
dΩ
Vol
)
❲r .∗ s t ♦♠♣① ♦♥t Re st♥s ♦r r ♣rt ♥ Vol ♦r t ♦
♠♥ ♦♠ s t s♣t♠♣♦r r ♦ t s②st♠ r♥ s ♥ s
❭t④②s♥♦r⑥ t t♦t ♥r② r s ♣♣r♦①♠t ② ♦♥② ♦♠♣t♥
t ♥t ♥r② r
〈Etot〉 =1
2Vol
(
ˆ
Ωx
ρω2wn(x, ω, φ).w∗n(x, ω, φ)dΩ)
r♦♣ ♦t② t♦rs Cgn(ω, φ) r ♦♠♣t ♦r ♥♠rs t rq♥②
st s ♦r t r♠ ♣②r♠ tt
♣r♦♣♦s ♠t♦♦♦② s s t♦ st② t s♣rs♦♥ ♥t♦ ♠♥s♦♥
r♠ ①t tt t ♦♥ssts ♦ ♥ ♥♥t ♣r♦ ♠♥s♦♥ ♠
♦ ♣r♦ strt♦♥ ♦ t ♥tr② ♣rs♥t ♥ r s②st♠ s ♠ ♦
2.5mm t ♣t ss♠② ♠ ♦ s♦tr♦♣ ♠♣ ♥♦♥♦♥ r ♣♦②♠r t t
s♠ ♠♥ rtrsts s ♦r t ♦♠♦♥st♦♥ ♦ ♦♥sr t ♥tr♥s
♠tr ♠♣♥ ♦ t ♦r ♠tr ♥ ②strt ♠♣♥ t♦r ♦ 0.001 s ♥ s
s♠ ♦♥sst♥t t♦ ②strt ♦ss t♦rs ♦ tr♠♦st♣♦①② ♣r♣rs ❬❪ s③
s 120mm2 ♠t♦ ♦s s t♦ ♦♠♣t ♥ rq♥s ♦rrs♣♦♥♥ t♦ ♥② k
t♦r sr ♥ ②♥r ♦♦r♥ts s②st♠ ② ts rs k ♥ ts ♥ φ ♥ t ♦
rst r♦♥ ♦♠♥ ♥♠r ♠♣♠♥tt♦♥ s s ♦♥ t ♦r♠
t♦♥ s♥ ♠♥s♦♥ ♦r♥tt♦♥ ♥ t ks♣ ② ♠♣♦s♥ Φ =
cos(φ)
sin(φ)
0
♣♣ ♦♥r② ♦♥t♦♥s r qts ♦ s♣♠♥ts ♦♥ t t♦ ♣rs
♦ tr s Γr+1Γr−
1andΓr+
2Γr−
2♦♥ ♦♥rs ♦ ♠♣♦s s rt ♦♥
r② ♦♥t♦♥s ♥ ①trs♦♥ ♦♣♥ r ♠♣s ♦ s♣♠♥ts r♦♠ t s♦r
s ①♣♦rt t♦ t st♥t♦♥ ♦rrs♣♦♥♥ t♦ t ♦♣♣♦st ♦♥ r♦♠ Γr+1toΓr−
1
s t ♦♠♥s r ♦ t s♠ s♣ ♠♥s♦♥ t②♣② s ♣♦♥ts ♠♣♣♥
①♣♦rt ♠♣♣♥ s s♦ ♦♣ t♦ t st♥t♦♥ s♣♠♥t ② s♥ t
r♥ ♠t♣rs ♠♣♠♥tt♦♥ s ♠ t t♣②ss ♣t♦r♠
♥ ♣r♠tr ♦♠♣tt♦♥ t♦ ♦t♥ k(ω, φ) s rr ♦t t t r♦t♥s ♦r
♣r♠trs ω andφ t qrt ♥ ♣r♦♠ ♥ r♦r♠t s rst
♦rr ♦♥ ② ♦♥ t stt ♠♥s♦♥ tr ♦♥str♥t ♥♥ t s ♣♦ss t♦ rt
t s②st♠ ♥ t ♦r♠ Ax = λBx ♦rt♠ ♦♠♣ts t rst ♥
s ♦ t ♠tr① C = A−1B ♦ ♦ ts t s♦r ss t P
r♦t♥s ♦r rs ♥ ♣r♦♠s s sr ② ❬❪ s ♦ s s
♦♥ r♥t ♦ t r♥♦ ♦rt♠ t ♠♣t② rstrt r♥♦ ♠t♦
P r♦t♥s ♠st ♣r♦r♠ sr ♠tr①t♦r ♠t♣t♦♥s Cv
r ♦♠♣s r ② s♦♥ t ♥r s②st♠ Ax = Bv s♥ t P
s♦r ♦♣ ② ❬❪ s ♣r♦r ss ♦ ♣rs♦♥ ♦t♥ ♣♦♥t ♥♠rs ♥ s
♠♣♠♥t s♥ ♦t ♦ ♦r ♠♠♦r② ♠♥♠♥t ♥ ♦rr t♦ ♦ ♥② ♠♠♦r② ♣r♦♠
♥ ♥ ♥s ♥ ♦♥r ♠s s ♦♥sr r rst ♠s s ♦♥ssts
♦ ttrr r♥ qrt ♠♥ts ♦r rs ♦ r♦♠ ♥ t r♥
♦♥ ♦ ttrr r♥ qrt ♠♥ts ♦r rs ♦ r♦♠ ♥ t
tst ss ♦♥sr t♥ ts ♦r t rq♥② r♥ s ω = 2π.[1000 : 1000 : 10000]
t♥ ③ ♥ ③ ♥ φ = [0 : π20
: π2]
❯
♥ ♣r♦♣rts
r s♦s t rt♦♥ ♦ t ♥♣♥ P♦ss♦♥s rt♦ νxy q t♦ νyx ♦r s②♠♠tr②
rs♦♥s ♦r r♥t δ s t r②♥ ♥tr♥ ♥s β = 0.05 ①t ♥t
P♦ss♦♥s rt♦ ♦r ♦ t r♠ tt s ♥t ♦r t θ r♥ ♦♥sr ♣rt
r♦♠ r② s♠ θ < 2o ♥ r θ > 78o ♠ ♥s ♥t ♦♥rt♦♥s r t
♠♥ts l t♥ t♦ ♠♦st t ♦r③♦♥t ♥s r rt t♦ ♦♥rt♦♥s
r t ♠♥ts t♥ t♦ st♥ ♣rt rt ♥ ♦t ss t ♠♥ts l t♥ t♦
♦♠♥t ② ① ♦r♠t♦♥ ♠♥s♠s ♦♥trt s♥♥t② t♦ ♦r♥
♦♥ t P♦ss♦♥s rt♦ s ♥ ♥trs②♠♠tr ♦♥②♦♠ ss♠s t rs ♥
①r ♦r ❬❪ t♥ 10o ♥ 30o t P♦ss♦♥s rt♦ t♥s t♦ ts st
♠♥t ∼ −0.9 ♦r δ = 5.0 t♥s t♦ rs s s♦♦♥ s t strt♥ ts ♦
t ♠♥ts strt t♦ ♣rs♥t ♥ t ♠r♦strtr ♠♦r t ♠♥t s♦ ss♠s
♣t s♣ ♦r♥ δ s t ♦r t ♠♥t ♦ t ♥♣♥ P♦ss♦♥s
rt♦ t t ♠♥♠♠ ♥♦ r t ♦r ♥tr♥ ♥ νxy = −0.6 ♦r
δ = 1.0 ♥ t ♦♥trr② t ♦t♦♣♥ P♦ss♦♥s rt♦ νxz r s ②s ♣♦st
r♥♥ t♥ ♥ t s♠♠♦♥♦t♦♥ rs t t ♥tr♥ ♥
♠♦r ♣r♦♥♦♥ t ♣t rs l δ = 1.0
r s♦s t rt♦♥ ♦ t ♥♣♥ Ex = Ey ♥ ♦t♦♣♥ Ez ❨♦♥s
♠♦s rss t ♥tr♥ ♥ ♦r t♦ r s♣t rt♦s δ = 1.0 ♥ δ = 3.0 ♥
♠♣♦rt♥t s♣t t♦ t s t t tt t tr♥srs st♥ss s ♦r t♥ ♥ t
♥♣♥ ♦♥ ♦r r② s♠ ♥s ♥st r♠ ♦ r t rt♦ Ex/Ez s ♦r
δ = 1.0 ♦r θ > 20o t rt♦ ♥ r② t♥ ♥ t♠s ♦r st♥ss rt♦s r
♦sr ♦r ♥rs♥ δ s s ♦r s ♥s ♦r ss r strtrs
♦ s♦ ♥r s♥ rss β ♦ t tr♥srs ♠♥ ♣r♦♣rts t ♥
♣♥ ♦♥s s t β3 ❬❪ ♦r t r♠ ♣②r♠ tt ♥ t♦t s
♣rs♠t ♦♥②♦♠ st♦♥ r t rs ♦r♠ ♥r ♥♥ t xy♣♥
♣r♦♣rts r ♦♠♥t ② ♠♠r♥ ♥ sr ♦r♠t♦♥s ♦ t rs l ♥ tt s♥s
t ♣②r♠ tt ♦♥sr ♥ ts ♦r s ♥trs②♠♠tr ♦♥②♦♠
t ts ♣rs♠t r♦ssst♦♥ s s tr♦tt♥ss r♥♦r♠♥t ♥ ♠♦r
♣r♦♥♦♥ rt♦ t♥ ♥♣♥ ♥ ♦t♦♣♥ sr ♥ ♦sr ♥ r
r Gxy/Gxz rt♦s ♣ t♦ ♥ ♦sr ♦r θ < 3o ♥ δ = 1.0 ♥trst♥② ♦r
♦t δ ♦♥rt♦♥s ♦♥ ♥ ♦sr tr♥srs sr st♥ss ♥rs♦♥ ♦ θ = 55o
r t ♦t♦♣♥ sr ♠♦s ♦♠s r t♥ t ♥♣♥ ♦♥ s ♦r
s ② t♦ ♦♥t t♦ ♥rs♥ ♥trt♦♥ t♥ t ♠♠r♥ st♥ss t♥
t rs l ♥ t ♥♥ ♦♥ ♥ t ♣ts b t t s ♦ t ♥t r
t♥s t♦ qs ♦r s♣ ♥ ♥ δ t ♦♥st♥t α
♦♥ss rs
r s♦s t s♦♥ss rs ♥ t xz ♣♥ q ♦r s②♠♠tr② ♥ t yz ♣♥
♦r δ = 1.0 r ♥ δ = 5.0 r t ♦♥st♥t ♥tr♥ ♥ θ =
20o t tr s♦♥ss r s3 (φ) s♦s r② str♦♥ rtt② t φ = 0, π/2, π, 3π/2
♦rrs♣♦♥♥ s♦ t♦ t ♠♥ ①s ♦ t rs l rst t♦ s♦♥ss rs s1 (φ) ♥
s2 (φ) ♦r ♠♦r ♥trst♥ ♦r t ♥r ③r♦s♦♣ t♥ 55o <
φ < 135o ③r♦s♦♥ss s ♦r ♥ tr♠s ♦ r♥ ♦r t s♦♥ s♦♥ss r ♥
♣♥s ♦♥ t r s♣t rt♦ δ ❲ r♥ ♥ tr♠s ♦ ♥ φ r♥ ♦r t
③r♦s♦♣ ①sts ♦r t tt t δ = 1.0 t s1 (φ) r s♦s t s♦♣ ①t♥♥
♠♦st t♥ 20o < φ < 160o ♦r rs l ss♠♥ ♠♦r ♠ strtr δ = 5.0
♥ ♥♦r♠s tr♠s t s♦♥ s♦♥ss r ♥ δ = 1.0 ♦rrs♣♦♥s t♦ r ♦ts
v s♦t♦♥s ♦ t rst♦s qt♦♥ t♥ t ♦♥s ♦r t tt ♦♥rt♦♥
t δ = 5.0 ①st♥ ♦ ③r♦s♦♣ s♦♥ss rs ♥ ♥trs②♠♠tr ♠♦♥♦♥
tts s ♥ sst s ♣♦ss ♠♥s♠ t♦ ♥rt s♦t♦♥s ♦♥ s♣ s♣t
rt♦♥s ♦♥ t ♠r♦strtr ❬❪ ♥ tt s♥s t r♠ ♣②r♠ ①t tt
s♦ s♦♠ ♥trst♥ ③r♦s♦♣ rtrsts ♦r t rst ♥ s♦♥ s♦♥ss rs
t♦ ♥♦t ♦r t rst ♦♥ s ♥t ♥ s♣ ♠♦♥♦♥ ♦♥rt♦♥s ② ❲♥
♥ ②♥ ❬❪ ♦ ③r♦s♦♣ s♦♥ss r ♥ ♦sr ♦r t ♣r♦♣t♦♥ ♥
t xy ♣♥ t ♥ s♦tr♦♣ rtt② ♦r s1 (φ) ♥ s3 (φ) ♥ str♦♥ rt♦♥t② ♥
s2 (φ) ♦r φ = 0, π/2, π, 3π/2 r s ♥ t s ♦ t xz ♣♥ ♣r♦♣t♦♥
t ♦ts ♦rrs♣♦♥♥ t♦ t s♦♥ s♦♥ss r t♥ t♦ t♠s ♦r ♥
♥♦r♠s tr♠s ♦r t ♠ tt rs δ = 5.0 t♥ t q s♣t rt♦ ♥
♣t ♦♥rt♦♥s t δ = 1.0
❲ s♣rs♦♥
♥ ts st♦♥ q♥tts ♦ ♥trst r ♥ ♥ ♥♦♥♠♥s♦♥ ♥ts rr♥
♦t② s ♦♥sr s
cT =
√
Ec
2(1− νc)ρc
r Ec νc ♥ ρc r t ❨♦♥s ♠♦s P♦ss♦♥s rt♦ ♥ ♥st② ♦ t s ♠tr
♥ t ss♦t rr♥ rq♥② s
fref =πcTL
s♣rs♦♥ ♦♥ Γ−X rt♦♥ ♦ t s②st♠
r s♦s t r ♣rt ♦ t s♣rs♦♥ rs Re(kn) ♥ t s ♦ ♥r♥
♠ss ♥♥ t s ♥♠r ♦t♥ rsts ♥t r ♦♠
♣①t② ♦ t r♦♦st ♦r ♦ t r♠ ♣②r♠ tt s②st♠ ♥♦r♣♦rt♥
♥♠r♦s ♥s♥t ♠♦s ♥ ♦♠♣① s♣rs♦♥ rs ♦r ♣r♦♣t ♦♥s s st
♠♦ s t ② s♥ ♥r② ♦t② rtr♦♥ s♦ tt Cgn(ω, φ).Φ > τcT
t τ = 0.5% rt♥ ♣r♦♣t s r ♣rs♥t ♥ r t s ♣♦ss
t♦ t t s②st♠ rq♥② ♥ ♣s t t rst ♦♥ t♥ ♥ ♦
t rr♥ rq♥② t s♦♥ t♥ ♥ ♥rr♦ ♥♣ ♦♥ ♥
♥t r♦♥ ③ t st ♦♥ s ♣rs♥t t♥ ♥ t♥r
♠ s ♠♦s A0 So ♥ ♦sr ♥ r② ♦ rq♥② ♥ ♦rr
s♣♦♥♥ r♦♣ ♦ts r♠ s s♦♥ ♥ r r ♥ ♣ strtr ♥
♣rtr ♣r♦♣t rtrsts ② ttrt t♦ t ♥s ①t ♦r ♥
♦sr t♥ ♥ ♦ t rr♥ rq♥②
s♣rs♦♥ ♦ t s②st♠ ♥ t ♦ ♠♥s♦♥ s♣
♣r♦♣♦s ♦♠♣tt♦♥ ♠t♦ ♦s s t♦ ♦♠♣t ♠t ♠♦ s ♣r♦♣
t♦♥ ♥ t ♦♠♣t ♠♥s♦♥ ks♣ ♦ t rst r♦♥ ③♦♥ ♣r♦♣♦s
♠t♦♦♦② s s ♦♥ t ♦♠♣tt♦♥ ♦ rq♥②♣♥♥t ♦♠♣① ♥♠rs
♦ t♦r♠ s ①♣♥ ♥ t s ♦ ♠♣ s②st♠s ♥ t rsts ♦t♥
♦♠ ♦♠♣① ♥trt♥ ♣s ♦t② ♥ ♥s♥t ♣rt ♦r ♦♠♣t
♥♠r ss♦t t♦ t r ♥ ♠♥r② ♣rts ♦ t ♦t♥ ♥s ♦ qt♦♥
♠♣♥ ♦r s ♥tr♦ ② ss♠♥ ♦♠♣① ♦♦ stt② t♥s♦r
s♠ ♠t♦♦♦② ♦ ♥ r③ ② ♥tr♦♥ ♥② ♥ ♦ ♥r s♦st
♠♦♥ s s s♦s ♦r ♦r ♥② ♦tr ♦♠♣① rq♥②♣♥♥t tr♠s
r strts t t②♣ rsts ♦ t ♥②ss Pr♦♣t ♥♠rs ♦ t
♠♣ s②st♠ r s♦♥ ♦r r♥t ♥s r♦♠ φ = 0 t♦ φ = 45o t st♣ ♦ 9o t
♥ ♦sr tt φ = 0 t s②♠♠tr② strt ♥ r st ①sts s
s♦♦♥ s ♦tr rt♦♥s r ♦♥sr t ♣♣rs t t♦ rs♣ t ♦t♥ rsts
t t rt ♦t♦♥ ♥t♦ t rst r♦♥ ③♦♥ s ♥ ①♣♥ ② t t tt t
♣r♦t② ♦ t ♥t ♣ttr♥ s ♦st ♥ t ♦r♥tt♦♥ s ♥♦t ♣r t♦ ♦♥ ♦ t ss
♦ t ♥t ♥ s t♦ ♠♥s♦♥ ♦♥sst♥t tr♥s♦r♠t♦♥ ♦sr ♥
♣ strtr s♦ ♣♥s ♦♥t♦ t ♣r♦♣t ♥s r♦r t♦ t t ♥♣
♦ t ♣r♦ s②st♠ ♥ ♥t♦r ♦ ♠♥♠ ♥s♥ rt♦ ♦ t ♦♠♣t s
♦r ♦♥sr rq♥② ♥ s ♥ s
Ind(ω, φ) = minn
∣
∣
∣
∣
Real(λn)
|(λn)|
∣
∣
∣
∣
r s♦s t ♣♦t ♦ ts ♥t♦r ♥ t ♦ ♦♠♥ ♦t♦♥ ♦ t
♥ ♣s ♥ ♦sr r② ♥ ♥t tt s♣ rtt② ♥ ①sts t rt♥
rq♥s s t ♥ ♣s r ♥♦t ①s②♠♠tr ♥ ♥ ♦t t♥ s♣
♥s ①♠♠ ♥s♥ s ♦sr ♦♥ t rt♦♥s ♦ t r♠ ♣②r♠
rs ♦ ♥t l ♦r ♦ rq♥② r♥s ♦ ③ t♥ ♥ ③ t
♠①♠♠ ♥s♥ s♦s ♥ s♦tr♦♣ rtt② ♦ ③ t ♠①♠♠
rtt② s ♥ ♦sr ♦♥ t ①s ♦ t s♠♥ts l ♦ t ①t tt
❱ ❯
r♠ ♣②r♠ tt sr ♥ ts ♦r s s♦♥ t ♦♦♥ r
trsts
tt s ♥ ♥♣♥ ①t ♦r t s ♦ t t P♦ss♦♥s
rt♦ ♣♥♥ ♦r t ♦♠tr② ♣r♠trs ♦ t r ♦♥rt♦♥s Pt
rs t♥ t♦ rs t ①tt② ♦ t ♦♥②♦♠ ♦t♦♣♥ P♦ss♦♥s
rt♦ s ♣♦st ♦r t ♦♥rt♦♥s ♦♥sr
♥♣♥ ♥① st♥ss ❨♦♥s ♠♦s Ex s r ♦♠♣r t♦ t tr♥s
rs s r♠ ♣②r♠ tt s ♣rs♠t ♦♥②♦♠ t
t ♦r♥t tr♦ t t♥ss tr♦r ①♣♥♥ ts rtr ♣r
♠♥ ♦r s♠r tr♥ s ♦sr s♦ ♦r t sr ♠♦s
t♦ t s ♣♦ss t♦ ♥t② ♦♠♥t♦♥ ♦ ♥tr♥ ♥ ♥ r s♣t
rt♦ δ tt s t♦ Gxy = Gxz stt♦♥
❩r♦s♦♣ s♦♥ss rs ♥ ♦sr ♦r t ♣r♦♣t♦♥ ♥ t xz ♣♥
♦r t rst ♥ s♦♥ rs rs♣t② sst♥ ♣♦ss rt♦♥s ♦r t
t♦r r s♦t♦♥s ♦ rt s♥ ts tt s ♣t♦r♠ ♦ ③r♦rtr
♦r t s♦♥ss rs s ♥ ♥t ♥ t xy ♣♥
❲♥ ♦♥sr♥ ♠♥s♦♥ ♣r♦♣t♦♥ t♥ ♥t♦ ♦♥t t t
s ♦ t ♠r♦strtr ♥♦t ♥ ♦♥♥ t♦ t ♦♥ ♥t ♣♣r♦①♠
t♦♥ ♦ ♣♦♥ts t r♠ tt s♦s ♦♠♣① ♣ttr♥ ♥ ♥ t xy
♣♥ ♥ s♠ s ♦ ②strt ♠♣♥ s ♦♥sr ❯s♥ ♥ ♥s♥
♥① rt ♦r ts tt t s ♣♦ss t♦ ♥t② ♥s♥ rtt② ♣t
tr♥s t rt♦♥t② ♣♥♥ ♦ t ♠t♠♦ rtrsts ♦ t r♠
①t ♣②r♠ tt
♥♦♠♥ts
s ♦r s ♥ ♣rt② s♣♣♦rt ② r♥t♥ ♦r P ♦s ② t ♣♥
♦t② ♦r Pr♦♠♦t♦♥ ♦ ♥
❬❪ t ❲② P②r♠ tt trss strtrs t ♦♦ trsss t
rs ♥ ♥ ♥♥r♥ ♦❭♥♦④♦⑥④
♠s⑥
❬❪ t ❲② t♥♠ ♦② tt trss strtrs trs ♠♣
s♥ ♦❭♥♦④♦⑥④♠ts
⑥
❬❪ ❲② t♥t♦♥ ♣r♦ r ♠ts P♦s♦♣ r♥st♦♥s ♦ t ♦②
♦t② t♠t P②s ♥ ♥♥r♥ ♥s ♦
❭♥♦④♦⑥④rst⑥
❬❪ P♥ ❱ s♣♥ ❲② ♦♣s ♠♥s♠ ♠♣s ♦r
♦♦ ♣②r♠ tt Pr♦♥s ♦ t ♦② ♦t② t♠t P②s ♥
♥♥r♥ ♥ ♦❭♥♦④♦⑥④rs♣⑥
❬❪ rs♦♥ rs♦♥ ①t trs Pr♦♥s ♦ t ♥sttt♦♥ ♦ ♥
♥♥rs Prt ♦r♥ ♦ r♦s♣ ♥♥r♥
❬❪ ♦♥ ③③♥ ♥②ss ♦ ♥♣♥ ♣r♦♣t♦♥ ♥ ①♦♥ ♥ r♥tr♥t
tts ♦♥ ❱rt
❬❪ ♣♦♥ ③③♥ ♠r ♥ ①♣r♠♥t ♥②ss ♦ tt ♦♠♣♥ ♦
r rss ♦r r♦s ♦r♥ ♦ ♥s ♦ trs ♥ trtrs
❬❪ ♣♦♥ r♣ ③③♥ ❲ ♣r♦♣t♦♥ ♥ ①t ttrr ♦♥②
♦♠s ♦r♥ ♦ ❱rt♦♥ ♥ ♦sts
❬❪ rt♥ ②rr♥r ♠t r♣ P♦ttr ③③♥ ①r
♣rs♠t ♥♦① ♦♥♣t P②s tts ♦
❬❪ tt♥ P r♦ ♥r♦ ③③♥ ♣♦♥ ♦♠♣♦st r str
trs ♦r ♠♦r♣♥ r♦s ♠r ♥②ss ♥ ♦♣♠♥t ♦ ♠♥tr♥ ♣r♦ss
♦♠♣♦sts Prt ♥♥r♥
❬❪ r♠ ❲♠s tt ♥s ♦♥ ♦ ①t ♥t♦r ♣♦②♠rs
t r♦♠ ①❬❪r♥ ♥ ♦s ♦r ♠t♦♥ ♦
❭♥♦④♦⑥④⑥
❬❪ ❱ ♦ ♦ ♦③♦ ❩♥ ♥ts
♠♥ ♥ ♥ ♦ P♦ss♦♥s t♦ ♦r r♦♥ ♥♦t ts ♥
♦❭♥♦④♦⑥④s♥⑥
❬❪ ❱ ♦ ♦③♦ ❩♥ ♥ts ã♦ ♠♥
♦♥ t ①t tr♥st♦♥ ♦r r♦♥ ♥♥♦t sts P②s
♦❭♥♦④♦⑥④P②s⑥
❬❪ ♦♠ t♦ ♦♣♠♥t ♦ ② s♥ ❯trt ♦r trtrs
♥tr♥t♦♥ ♦r♥ rs
❬❪ t♦ r♣ r♠ ♦♠♣♦st ①t ♦♥②♦♠ ♥ t ♥tr♥t♦♥
♦♥r♥ ♦♥ ♦♠♣♦st trtrs
❬❪ t♦ ♥s r♣ r r♠ ♦r♣♥ ❲♥♦① ♦♥♣t ♦r♥ ♦
♥t♥t tr ②st♠s ♥ trtrs ♦❭♥♦④♦⑥④
❳⑥
❬❪ P ❲♦ ♠♥ P♦♥♦♥s ♠r ❯♥rst② Prss ♠r ❯
❬❪ ♦r♥rt rt P ♦r♠♥ ♦♠♦é♥ést♦♥ ♥ ♠é♥q s ♠tér①
♥ r♦♣ t
❬❪ r ♦♥sttt ♠♦♥ ♦ ♣③♦tr ♣♦②♠r ♦♠♣♦sts t tr
♦❭♥♦④♦⑥④t♠t
⑥
❬❪ ② ❲ ♣r♦♣t♦♥ ♥ ②r ♥s♦tr♦♣ ♠ t ♣♣t♦♥s t♦ ♦♠♣♦sts
♦rt ♦♥ sr ♥ ❱
❬❪ ❲♥ ②♥ ①st♥ ♦ ①tr♦r♥r② ③r♦rtr s♦♥ss r ♥ ♥s♦tr♦♣
st ♠ ♦r♥ ♦ t ♦st ♦t② ♦ ♠r
♦❭♥♦④♦⑥④⑥
❬❪ r ♦♥s ♦ ❲s ♦♠♦♥③t♦♥ ♥ ♣tr s②♠♣t♦t ♥②ss ♦r
♥ té♠tqs Prs t ♣♣qés
❬❪ ♦ r♦t ♥ s r♦♣ ♥ ♥r② ♦ts ♥t
♠♥ts ♦r♥ ♦ ♦♥ ♥ ❱rt♦♥ ❳ ♦
❭♥♦④♦⑥④④s⑥⑥
❬❪ ♥ ♦ t♠♦ ♣r♦♣t♦♥ ♥ s♦♥ ♥ strtrs tr♦ ♥t
♠♥ts r♦♣♥ ♦r♥ ♦ ♥s ♦s
♦❭♥♦④♦⑥④④r♦♠s♦⑥⑥
❬❪ ♦♦♥ ♦ ③q ❲ ♠♦t♦♥ ♥ t♥ strtrs ♦r♥ ♦ ♦♥
♥ ❱rt♦♥ ❳ ♦❭♥♦④♦⑥④④s
⑥⑥
❬❪ ♦t ss ③③♥ ♦ ♦qt♦ ♦♠♣♦st♦♥ ♦ t st♦
②♥♠ qt♦♥s ♣♣t♦♥ t♦ ♠♥s♦♥ s s♣rs♦♥ ♦♠♣tt♦♥ ♦ ♠♣
♠♥ s②st♠ ♥tr♥t♦♥ ♦r♥ ♦ ♦s ♥ trtrs
❬❪ ♦t ♥r ♦ ❲ ♦t♦♥ ♣t♠③t♦♥ ♥ Pr♦② strt
♥t P③♦♦♠♣♦st ♠ trtrs ♦r♥ ♦ ♥t t ②st ♥ trt
❬❪ r ♦♠♣tt♦♥ ♦ ♣r♦♣t s ♥ r r s♥ ♥t ♠♥t t♥q
♦r♥ ♦ ♦♥ ♥ ❱rt♦♥
❬❪ ❲ ②s♥ör ör♣rs♥r r♥♥ ③r r♥♥ ♦♥ ♥rt♥ ♥
♥t♥stät♥ ❲ss♥st ❱rssst tttrt
❬❪ rt♦t ssrr ❨ r♥ ♠♣♥ ♥②ss ♦ ♦♠♣♦st ♠
trs ♥ strtrs ♦♠♣♦st trtrs ♦
❭♥♦④♦⑥④♦♠♣strt⑥
❬❪ ♦q ♦r♥s♥ ❨♥ P srs s♦t♦♥ ♦ rs ♥
♣r♦♠s t ♠♣t② rstrt r♥♦ ♠t♦s ♠
❬❪ ♥ ärt♥r ♦♥ ♥s②♠♠tr s♣rs s②st♠s ♦ ♥r qt♦♥s t P
tr ♥rt♦♥ ♦♠♣tr ②st♠s ❳
❬❪ r♣ P P♥②♦t♦ ♦♠♥s♦♥ ♠r ♥ ①♣r♠♥t ♥① ♦♥ ♦♥ ♥
♣♥ ①t ♦♥②♦♠s ♦r♥ ♦ tr♥ ♥②ss ♦r ♥♥r♥ s♥
♦❭♥♦④♦⑥④⑥
❬❪ s♦♥ s② ♥s ♦ r♠♥s♦♥ r trs Pr♦
♥s ♦ t ♦② ♦t② ♦ ♦♥♦♥ t♠t ♥ P②s ♥s
♦❭♥♦④♦⑥④rs♣⑥
r ❱ ♣ t ♣②r♠ r♠ ♦r t ts ♥♦♥♠♥s♦♥ ♦♠tr②
♣r♠trs ❱ t α = 1, β = 0.05, δ = 6, θ = 20o
r ♥r ♣r♦ s
r ②♦t ♦ t ♠♣ r♠ tt ♥t ♦♥sr ♦r t
♣r♦♣t♦♥ st②
r ❯♥r♥ ♥ r♥ ♠s ss
0 10 20 30 40 50 60 70 80−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
θ [o]
ν xy
δ = 1.0
δ = 3.0
δ = 5.0
0 10 20 30 40 50 60 70 800.2
0.25
0.3
0.35
0.4
0.45
0.5
θ [o]
ν xz
δ = 1.0
δ = 3.0
δ = 5.0
r strt♦♥ ♦ t P♦ss♦♥s rt♦s νxy ♥ νxz rss t ♥ θ ♦r
r♥t δ ♣r♠trs ♥ α = 1
20 30 40 50 600
0.5
1
1.5
2
2.5
3
3.5
4
4.5
θ [o]
2 4 6 80
50
100
150
200
θ
Ex/E
c X 103, δ = 1.0
Ez/E
c X 103, δ = 1.0
Ex/E
c X 103, δ = 3.0
Ez/E
c X 103, δ = 3.0
30 40 50 60 70 800
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
θ [o]
2 4 6 8 100
10
20
30
40
θG
xy/G
c X 103, δ = 1.0
Gxz
/Gc X 103, δ = 1.0
Gxy
/Gc X 103, δ = 3.0
Gxz
/Gc X 103, δ = 3.0
r ♦♥♠♥s♦♥ ♥♣♥ ♥ tr♥srs ❨♦♥s ♠♦s rss t ♥tr♥
♥ θ ♦r r♥t s♣t rt♦s δ ♠r ♣r♠tr rs ♦r ♥♦♥♠♥s♦♥
♥♣♥ sr ♥ tr♥srs ♠♦s ♦r t♦♥s α = 1
0.5
1
1.5
30
210
60
240
90
270
120
300
150
330
180 0
θ = 20o
s1(φ)
s2(φ)
s3(φ)
δ = 1.0
θ = 20o
s1(φ)
s2(φ)
s3(φ)
δ = 5.0
r ♦♥ss r ♥ t xz ♣♥ ♦r α = 1, θ = 20o
0.1 0.2 0.3
θ = 20o
s1(φ)
s2(φ)
s3(φ)
δ = 1.0
θ = 20o
s1(φ)
s2(φ)
s3(φ)
δ = 5.0
r ♦♥ss rs ♥ t xy ♣♥ ♦r α = 1.0, θ = 20o
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Real part of reduced wave number
Re
du
ce
d f
req
ue
ncy
r s♣rs♦♥ rs ♦ ♠♦s ♦ t st s②st♠ ♠♥r② ♣rt ♦ λn(ω)
♦r r ♣rt ♦ kn(ω)
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Real part of reduced wave number
Re
du
ce
d f
req
ue
ncy
r s♣rs♦♥ rs ♦ ♣r♦♣t ♠♦s ♦ t st s②st♠ ♠♥r② ♣rt
♦ λn(ω) ♦r r ♣rt ♦ kn(ω)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.140
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Real part of reduced group velocity
Re
du
ce
d f
req
ue
ncy
r s♣rs♦♥ rs ♦ ♣r♦♣t ♠♦s ♦ t st s②st♠ ♠♥r② ♣rt
♦ λn(ω) ♦r r ♣rt ♦ kn(ω)
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
Reduced fre
quency
φ=0
0 0.5 1 1.50
0.02
0.04
0.06
0.08
0.1
φ=π/20
0 0.5 1 1.50
0.02
0.04
0.06
0.08
0.1
φ=π/10
0 0.5 1 1.50
0.02
0.04
0.06
0.08
0.1
Reduced fre
quency
Propagative part of reduced wave number
φ=3π/20
0 0.5 1 1.50
0.02
0.04
0.06
0.08
0.1
Propagative part of reduced wave number
φ=π/5
0 0.5 1 1.50
0.02
0.04
0.06
0.08
0.1
Propagative part of reduced wave number
φ=π/4
r Pr♦♣t ♥♠rs ♦ ♠♣ s②st♠ Im(λn(ω) ♦r r♥t ♥s
r rtt② s♥ t ♥s♥ ♥① strt t ♥t