Characterization of Elasticity-Tensor Symmetries Using SU(2)

Abstract

A symmetry class of an elasticity tensor, c, is determined by the variance of this tensor with respect to a subgroup of the special orthogonal group, SO(3). Using the double covering of SO(3) by the special unitary group, SU(2), we determine the subgroups of SU(2) that correspond to each of the eight symmetry classes. A family of maps between C2 and R3 that preserve the action of the two groups is constructed. Using one of these maps and three associated polynomials, we derive new methods for characterizing the symmetry classes of elasticity tensors.

Full text

SU (2)
c
SO(3) S O(3) S U (2)
SU (2)
C2R3
O(3)
−I3R3
SO(3)
SU (2)
SO(3) SU (2)
SU (2)
C2R3
SO(3) SU (2)
SO(3)
SO(3)


SU (2)
SU (2)
SU (2)
SU (2)
SO (3)
R3
c
c:R3×R3×R3×R3−→ R
c(u, v, z, w) = c(v , u, z, w) = c(z, w, u, v),∀u, v, z, w ∈R3.
{e1, e2, e3}R3
cijkl =c(ei, ej, ek, el), i, j, k, l ∈ {1,2,3}.
c34= 81 cijkl
c
c(u, v, z, w) = cij kl uivjzkwl,

SU (2)
u=uieiv=vieiz=zieiw=wiei
cijkl =cj ikl =cklij,
i, j, k, l ∈ {1,2,3}
c
1c1111,1c2222,1c3333 ,
4c2323,2c2233,4c1212 ,2c1122,4c1313 ,2c1133 ,
4c1123,8c1213,4c1233 ,8c1323,4c2213 ,8c1223 ,
4c1222,4c1112,4c2223 ,4c1113,4c2333 ,4c1333 .
c
c
{e1, e2, e3}
Vij =ci1j1+ci2j2+ci3j3,
Dij =c11ij +c22ij +c33ij.
V D
SU (2)
O(3)

c
A
c(Au, Av, Aw, Az ) = c(u, v, w, z),∀u, v, z, w ∈R3.
Gc
O(3)
Gc−I3
A
−A A −A
SO(3)
e
GcGc
e
Gc
f
Gc=Gc∩SO(3) Gc=−f
Gc∪f
Gc.
SO(3) SU (2)
SU (2)
SO(3) SU (2) ψ:SU (2) −→ SO(3)
A∈SO(3)
±B∈SU (2) ψ(±B) = A
ψ
ψ
R3
C2ψ
ψ:±cos θ/2ιsin θ/2
ιsin θ/2 cos θ/27→ 

1 0 0
0 cos θ−sin θ
0 sin θcos θ
,
ψ:±cos θ/2 sin θ/2
−sin θ/2 cos θ/27→ 

cos θ0−sin θ
0 1 0
sin θ0 cos θ
,
ψ:± eιθ/20
0e−ιθ/2!7→ 

cos θ−sin θ0
sin θcos θ0
0 0 1 
,
ι=√−1
ψ

SU (2)
SU (2)
Hcψ(Hc) = e
GcHc
Gce
Gc
Gc={±I3}
e
Gc={I3}Hc={±I2}
R3
Gc={±I3,±Re3}Re3
e3
e
Gc={I3,−Re3}SU (2)
Hc=±I2,±ι0
0−ι.
{e1, e2, e3}R3e3
{e1, e2}
c1111, c2222 , c3333 ,
c1122, c1133 , c2233 ,
c1212, c1313 , c2323 ,
c1112, c1222 , c1233 , c1323 .
Gc={±I3,±Rei, i ∈ {1,2,3}}
e
Gc={I3,−Rei, i ∈ {1,2,3}}
SU (2)
Hc=±I2,±0ι
ι0,±0 1
−1 0 ,±ι0
0−ι.
{e1, e2, e3}R3e1, e2, e3
c1111, c2222 , c3333 ,
c1122, c1133 , c2233 ,
c1212, c1313 , c2323 .
Gc={±I3,±Ruα,±R±2π/3,e3, α ∈ {1,2,3}}
Ruα, α ∈ {1,2,3}e3

2π/3
uα
e3e2uα
uα= sin(θα/2)e1−cos(θα/2)e2θα∈ {0,±2π/3}
e
Gc={I, −Ruα, R±2π/3,e3, α ∈
{1,2,3}} SU (2)
Hc=(±I2,±0 1
−1 0 ,± 0e±ιπ/3
−e∓ιπ/30!,± e±ιπ/30
0e∓ιπ/3!).
{e1, e2, e3}R3
c1111 =c2222, c3333,
c1122, c1133 =c2233,
c1212 =1
2(c1111 −c1122), c1313 =c2323,
c1123 =−c2223 =c1213.
Re3
e3
c1123 =−c2223 =c1213
Gc={±I3,±R±π/2,e3,±Rπ,e3,±Ruα, α ∈ {1,2,3,4}}
Ruα, α ∈ {1,2,3,4}
e3π/4
uα
e3e1e2
uα= sin(θα/2)e1−cos(θα/2)e2θα∈ {0,±π/2, π}
e
Gc={I3, R±π/2,e3, Rπ,e3,−Ruα, α ∈
{1,2,3,4}} SU (2)
Hc=(±I2,± e±ιπ/40
0e∓ιπ/4!,±ι0
0−ι,±0ι
ι0,
±0 1
−1 0 ,± 0ιe±ιπ/4
−e∓ιπ/40!)
{e1, e2, e3}R3
c1111 =c2222, c3333 ,
c1122, c1133 =c2233,
c1212, c1313 =c2323.

SU (2)
Gg=

±

cos θ−sin θ0
sin θcos θ0
0 0 1 
,±

cos θsin θ0
sin θ−cos θ0
0 0 1 
, θ ∈(−π, π]

.
±Rθ,e3
θ e3±Ru
u(θ) = sin(θ/2)e1−
cos(θ/2)e2
e3
e3
O(2) O(3)
e
Gc={Rθ,e3,−Ru(θ), θ ∈
(−π, π]}SU (2)
Hc=(± eιθ/20
0e−ιθ/2!,± 0eιθ/2
−e−ιθ/20!, θ ∈(−π, π]).
{e1, e2, e3}R3e3
c1111 =c2222, c3333,
c1122, c1133 =c2233,
c1212 =1
2(c1111 −c1122), c1313 =c2323.
Gc={A∈O(3), A(ei) = ±ej, i, j ∈ {1,2,3}}.
e
Gc={A∈SO(3), A(ei) = ±ej, i, j ∈

{1,2,3}} SU (2)
Hc=±I2,±0ι
ι0,±0 1
−1 0 ,±ι0
0−ι,
± e±ιπ/40
0e∓ιπ/4!,± 0e±ιπ/4
−e∓ιπ/40!,
±cos π/4±ιsin π/4
±ιsin π/4 cos π/4,±cos π/4∓sin π/4
±sin π/4 cos π/4,
±∓sin π/4ιcos π/4
ιcos π/4∓sin π/4,±±ιsin π/4 cos π/4
−cos π/4∓ιsin π/4,
±ιcos π/4∓sin π/4
±sin π/4−ιcos π/4,±±ιsin π/4ιcos π/4
ιcos π/4∓ιsin π/4,
±±sin π/4 cos π/4
−cos π/4±sin π/4,±ιcos π/4∓ιsin π/4
∓ιsin π/4−ιcos π/4.
{e1, e2, e3}R3e1, e2, e3
c1111 =c2222 =c3333,
c1122 =c1133 =c2233,
c1212 =c1313 =c2323.
Gc=O(3)
e
Gc=SO(3) Hc=SU (2)
{e1, e2, e3}R3
c1111 =c2222 =c3333 = 2c1212 +c1122,
c1122 =c1133 =c2233,
c1212 =c1313 =c2323.
C2
SU (2)

SU (2)
C2R3
φ:C2→R3
SO (3) R3SU (2)
C2
C2−→ R3
φ
↓˜gA↓gψ(A)
φ
C2−→ R3
,
˜gAgψ(A)C2R3
A SU (2) ψ(A)SO(3)
ψ:SU (2) →SO (3)
gψ(A)(φ) = φ(˜gA).
(0, R)∈C2
r1, r2, r3∈R3(r1)2+ (r2)2+ (r3)2=R2R∈R
SU (2) SO (3)
C2R3z1, z2≡
p1+ιq1, p2+ιq2SU (2) ξ1, ξ 2, ξ3
SO (3) z1, z 2
SU (2) (0, R)∈C2z1, z 2
SO (3)
r1, r2, r3∈R3ξ1, ξ2, ξ3
z1, z2SU (2)
a b
−b a ,
|a|2+|b|2= 1 SU (2) (0, R)∈C2
n(Rb, Ra)∈C2,|a|2+|b|2= 1o.
z1, z2


z2
R
z1
R
−z1
R
z2
R




SU (2) (0, R)z1, z2
SU (2)
SO (3)
1
R z2z1
−z1z2!7→ 1
R2







Re (z2)2−(z1)2Im (z2)2−(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
−2Re z1z2−2Im z1z2z22−z12







.
r1, r2, r3∈R3
1
R2







Re (z2)2−(z1)2Im (z2)2−(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
−2Re z1z2−2Im z1z2z22−z12












r1
r2
r3




.
R φ
SU (2) SO (3)
φ:z1
z27→ 1
R2







Re (z2)2−(z1)2Im (z2)2−(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
−2Re z1z2−2Im z1z2z22−z12












r1
r2
r3




.
ξ1, ξ2, ξ3
C2
SO(3)
C2
SU (2)
cijkl
{e1, e2, e3}R3
P(ξ1, ξ2, ξ3) = cijklξiξjξkξl

SU (2)
PV(ξ1, ξ2, ξ3) = Vij ξiξj,
PD(ξ1, ξ2, ξ3) = Dij ξiξj,
V D
e
Gc
e
Gc={A∈SO(3), P (AX) = P(X), PV(AX) = PV(X), PD(AX) = P(X),∀X∈R3}.
C2
SU (2)
r1, r2, r3
φ:z1
z27→ 1
R2







Re (z2)2−(z1)2Im (z2)2−(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
−2Re z1z2−2Im z1z2z22−z12












r1
r2
r3





z1, z2r1, r2, r3=a(−1,−ι, 0)
a
R3
z1, z2a=
R2/2
ϕ: (z1, z2)7→ (ξ1(z1, z2) = (z1)2−(z2)2, ξ2(z1, z 2) = −ι((z1)2+(z2)2), ξ3(z1, z2) = 2z1z2)
P, PVPD
ϕ
P(z1, z2) = P((z1)2−(z2)2,−ι((z1)2+ (z2)2),2z1z2),

z1z2
PV(z1, z2) = PV((z1)2−(z2)2,−ι((z1)2+ (z2)2),2z1z2)
PD(z1, z2) = PD((z1)2−(z2)2,−ι((z1)2+ (z2)2),2z1z2),
z1
z2
ϕ SO(3) SU (2)
Hc
Hc={U∈SU (2), P (U Z ) = P(Z), PV(UZ) = PV(Z), PD(U Z) = P(Z),∀Z∈C2}.
P(8)(t) = P(z1/z2,1),
P(4)
V(t) = PV(z1/z2,1),
P(4)
D(t) = PD(z1/z2,1),
t=z1/z2
z1, z2˜
z1,˜
z2
Pz1, z2t=z1/z2
z2qP(q) z1
z2!=˜
z2qP(q) ˜
z1
˜
z2!,
q
P(8)
P(8)(t) = a8t8+a7t7+a6t6+a5t5+a4t4−a5t3+a6t2−a7t+a8,

SU (2)
a8=c1111 +c2222 −4c1212 −2c1122 + 4ι(−c1112 +c1222),
a7= 8[c1113 −c1322 −2c1223 +ι(−c1123 +c2223 −2c1213)],
a6= 4[−c1111 +c2222 −4c2323 + 4c1313 −2c2233 + 2c1133 + 2ι(c1112 +c1222 −2c1233 −4c1323)],
a5= 8[−3c1113 −c1322 + 4c1333 −2c1223 +ι(c1123 + 3c2223 −4c2333 + 2c1213)],
a4= 2(3c1111 + 3c2222 + 8c3333 −16c2323 −16c1313 + 4c1212 + 2c1122 −8c1133 −8c2233).
P(8)
P(8) c
c
P(4)
V
P(4)
V(t) = aV
4t4+aV
3t3+aV
2t2−aV
3t+aV
4.
aV
4=c1111 −c2222 +c1313 −c2323 −2ι(c1112 +c1222 +c1323),
aV
3= 4[c1113 +c1223 +c1333 −ι(c1213 +c2223 +c2333)],
aV
2= 2(−c1111 −c2222 + 2c3333 −2c1212 +c1313 +c2323).
P(4)
D
P(4)
D(t) = aD
4t4+aD
3t3+aD
2t2−aD
3t+aD
4.
aD
4=c1111 −c2222 +c1133 −c2233 −2ι(c1112 +c1222 +c1233),
aD
3= 4[c1113 +c1322 +c1333 −ι(c1123 +c2223 +c2333)],
aD
2= 2(−c1111 −c2222 + 2c3333 −2c1122 +c1133 +c2233).
P(4)
VP(4)
D

HcSU (2)
SU (2)
P(8) P(4)
VP(4)
D
Hc
P(8) P(4)
VP(4)
D
P(2q)(t) = εqP(2q)(t/ε), q ∈ {2,4},
ε2= 1 P(4) P(4)
VP(4)
D
P(z1, z2)PV(z1, z2)PD(z1, z2)
(z1, z2)7→ ±(z1, z2)
±(ιz1,−ιz2).
P(z1, z2)PV(z1, z2)PD(z1, z2)
P(8)(t)P(4)
V(t)P(4)
D(t)
t=z1/z27→ ±t=±z1/z2.
a7=a5=aV
3=aD
3= 0.
c1113 =c1223 =c1333 =c1213 =c2223 =c2333 =c1322 =c1123 = 0.

SU (2)
P(8) P(4)
VP(4)
D
P(2q)(t) = εqP(2q)(t/ε), q ∈ {2,4},
P(2q)(t) = εqt2qP(2q)(−ε/t), q ∈ {2,4},
ε2= 1 P(4) P(4)
VP(4)
D
P(z1, z2)PV(z1, z2)PD(z1, z2)
(z1, z2)7→ 






±(z1, z2)
±(ιz2, ιz1)
±(z2,−z1)
±(ιz1,−ιz2).
P(z1, z2)PV(z1, z2)PD(z1, z2)
P(8)(t)P(4)
V(t)P(4)
D(t)
t=z1/z27→ ±t=εt
±1/t =−ε/t.
Im(a8) = Im(a6) = Im(aV
4) = Im(aD
4) = 0.
c1112 =c1222 =c1233 =c1323 = 0.

P(8) P(4)
VP(4)
D
P(2q)(t) = ωqP(2q)(t/ω), q ∈ {2,4},
P(2q)(t) = ωqt2qP(2q)(−ω/t), q ∈ {2,4},
ω3= 1 P(4) P(4)
VP(4)
D
P(z1, z2)PV(z1, z2)PD(z1, z2)
(z1, z2)7→ 








±(z1, z2)
±(z2,−z1)
±(e±ιπ/3z2,−e∓ιπ/3z1)
±(e±ιπ/3z1, e∓ιπ/3z2).
P(z1, z2)PV(z1, z2)PD(z1, z2)
P(8)(t)P(4)
V(t)P(4)
D(t)
t=z1/z27→ ωt
−ω/t .
P(8)(ωt) = ωP (8) (t),
P(4)
V(ωt) = ω2P(4)
V(t),
P(4)
D(ωt) = ω2P(4)
D(t).
P(8)(t) = a7t7+a4t4−a7t=t(a7t6+a4t3−a7).
P(4)(t) = a2t2.
P(8) P(4)
V
P(4)
D
a8=a6=a5=aV
4=aV
3=aD
4=aD
3= 0.

SU (2)
ω=e2πι/3
q= 4 Re(a7) = 0
Re(a7) =
0
P(8) P(4)
VP(4)
D
P(2q)(t) = αqP(2q)(t/α), q ∈ {2,4},
P(2q)(t) = αqt2qP(2q)(−α/t), q ∈ {2,4},
α4= 1 P(4) P(4)
VP(4)
D
P(z1, z2)PV(z1, z2)PD(z1, z2)
(z1, z2)7→

















±(z1, z2)
±(e±ιπ/4z1, e∓ιπ/4z2)
±(ιz1,−ιz2)
±(ιz2, ιz1)
±(z2,−z1)
±(ιe±ιπ/4z2,−ιe∓ιπ/4z1).
P(z1, z2)PV(z1, z2)PD(z1, z2)
P(4)(t)P(4)
V(t)P(4)
D(t)
t=z1/z27→ αt
−α/t .
P(8)(αt) = P(8) (t),
P(4)
V(αt) = α2P(4)
V(t),
P(4)
D(αt) = α2P(4)
D(t).
P(8)(t) = a8t8+a4t4+a8.

P(4)(t) = a2t2.
α= 1 q= 4
Im(a8) = 0 P(8) P(4)
V
P(4)
D
Im(a8) = a7=a6=a5=aV
4=aV
3=aD
4=aD
3= 0.
Im(a8) = Re(a7) = a6=a5=aV
4=aV
3=aD
4=aD
3= 0.
Re(a8) = 0
Im(a7) = 0
P(8) P(4)
VP(4)
D
P(2q)(t) = zqP(2q)(t/z), q ∈ {2,4},
P(2q)(t) = zqt2qP(2q)(−z/t), q ∈ {2,4},
z=eιθ P(4)
P(4)
VP(4)
D
P(z1, z2)PV(z1, z2)PD(z1, z2)
(z1, z2)7→ (±(eιθ/2z1, e−ιθ/2z2)
±(eιθ/2z2,−e−ιθ/2z1).

SU (2)
P(z1, z2)PV(z1, z2)PD(z1, z2)
P(4)(t)P(4)
V(t)P(4)
D(t)
t=z1/z27→ (eιθ t
−eιθ/t .
P(4)
VP(4)
D
P(4)
V(t) = a(V)
2t2
P(4)
D(t) = a(D)
2t2
aV
4=aV
3=aD
4=aD
3= 0.
P(8) P(8)(t) =
a4t4
a8=a7=a6=a5= 0.
P(4)
VP(4)
DP(8)
P(2q)(t) = αqP(2q)(t/α),
P(2q)(t) = αqt2qP(2q)(−α/t),
2qP(2q)(t) = (t−α)2qP(2q)(β
a
t+α
t−α),
α4= 1 β2= 1
2qP(2q)(t) =
(P(z1, z2)PV(z1, z2)PD(z1, z2)

(z1, z2)7→



















































±(z1, z2)
±(ιz2, ιz1)
±(z2,−z1)
±(ιz1,−ιz2)
±(e±ιπ/4z1, e∓ιπ/4z2)
±(e±ιπ/4z2,−e∓ιπ/4z1)
±(cos(π/4)z1±ιsin(π/4)z2,±ιsin(π/4)z1+ cos(π/4)z2)
±(cos(π/4)z1∓sin(π/4)z2,±sin(π/4)z1+ cos(π/4)z2)
±(∓sin(π/4)z1+ιcos(π/4)z2, ι cos(π/4)z1∓sin(π/4)z2)
±(±ιsin(π/4)z1+ cos(π/4)z2,−cos(π/4)z1∓ιsin(π/4)z2)
±(ιcos(π/4)z1∓sin(π/4)z2,±sin(π/4)z1−ιcos(π/4)z2)
±(±ιsin(π/4)z1+ιcos(π/4)z2, ι cos(π/4)z1∓ιsin(π/4)z2)
±(±sin(π/4)z1+ cos(π/4)z2,−cos(π/4)z1±sin(π/4)z2)
±(ιcos(π/4)z1∓ιsin(π/4)z2,∓ιsin(π/4)z1−ιcos(π/4)z2).
PV(z1, z2)PD(z1, z2)
P(z1, z2)
P(8)(t)
P(4)
VP(4)
D
aV
4=aV
3=aV
2=aD
4=aD
3=aD
2= 0.
Im(a8) = a7=a6=a5= 0.
t= 1 α=−1
a4= 14a8.
P(8) P(4)
VP(4)
D
P(z1, z2)PV(z1, z2)PD(z1, z2)
P(8)
P(4)
VP(4)
D

SU (2)
P(8) P(4)
VP(4)
D
a8=a7=a6=a5=a4=aV
4=aV
3=aV
2=aD
4=aD
3=aD
2= 0.
P(2q)(t) = εqP(2q)(t/ε)ωqt2qP(2q)(−ω/t) 2−q(t−α)2qP(2q)(β
a
t+α
t−α)
ε2= 1
ε2= 1 ω2= 1
ε3= 1 ω3= 1
ε4= 1 ω4= 1
|ε|= 1 |ω|= 1
ε4= 1 ω4= 1 α4= 1, β2= 1
|ε|= 1 |ω|= 1 |α|= 1 |β|= 1
e3
e3
e1e2π/2
P(2q)(t)
P(2q)(t) = a0+a1t+···+a2qt2q,
a0=εqa0
a1=εq−1a1
an=εq−nan
a0=ω−qa2q

a1=−ω1−qa2q−1
an= (−1)nωn−qa2q−n.
ε2= 1 q
ε2= 1 ω2= 1
a2k=a2q−2k
ε3= 1 ω3= 1
ak
q−k
ak= (−1)kωk−qa2q−n
ε4= 1 ω4= 1
akq−k
ak= (−1)kωk−qa2q−n
|ε|= 1
n=q εn−q= 1
ε4= 1
ω4= 1
α4= 1 β2= 1
akq−k
ak= (−1)kωk−qa2q−n
t= 1
α=−1
an
n=q εn−q= 1

SU (2)
Tr. Isotropic 5,
8
Isotropic 2,
8
Orthotropic 9,8
Cubic 3,48
Monoclinic 13,4
Trigonal 6,12
Anisotropic 21,2
Tetragonal 6,16