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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.
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×R3R
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)
GcI3
A
A A A
SO(3)
e
GcGc
e
Gc
f
Gc=GcSO(3) Gc=f
Gcf
Gc.
SO(3) SU (2)
SU (2)
SO(3) SU (2) ψ:SU (2) SO(3)
ASO(3)
±BSU (2) ψ(±B) = A
ψ
ψ
R3
C2ψ
ψ:±cos θ/2ιsin θ/2
ιsin θ/2 cos θ/27→
1 0 0
0 cos θsin θ
0 sin θcos θ
,
ψ:±cos θ/2 sin θ/2
sin θ/2 cos θ/27→
cos θ0sin θ
0 1 0
sin θ0 cos θ
,
ψ:± eιθ/20
0eιθ/2!7→
cos θsin 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
0ι.
{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
0ι.
{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)e1cos(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
eιπ/30!,± e±ιπ/30
0eιπ/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)e1cos(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
0eιπ/4!,±ι0
0ι,±0ι
ι0,
±0 1
1 0 ,± 0ιe±ιπ/4
eιπ/40!)
{e1, e2, e3}R3
c1111 =c2222, c3333 ,
c1122, c1133 =c2233,
c1212, c1313 =c2323.
SU (2)
Gg=
±
cos θsin θ0
sin θcos θ0
0 0 1
,±
cos θsin θ0
sin θcos 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
0eιθ/2!,± 0eιθ/2
eιθ/20!, θ (π, π]).
{e1, e2, e3}R3e3
c1111 =c2222, c3333,
c1122, c1133 =c2233,
c1212 =1
2(c1111 c1122), c1313 =c2323.
Gc={AO(3), A(ei) = ±ej, i, j ∈ {1,2,3}}.
e
Gc={ASO(3), A(ei) = ±ej, i, j
{1,2,3}} SU (2)
Hc=±I2,±0ι
ι0,±0 1
1 0 ,±ι0
0ι,
± e±ιπ/40
0eιπ/4!,± 0e±ιπ/4
eιπ/40!,
±cos π/4±ιsin π/4
±ιsin π/4 cos π/4,±cos π/4sin π/4
±sin π/4 cos π/4,
±sin π/4ιcos π/4
ιcos π/4sin π/4,±±ιsin π/4 cos π/4
cos π/4ιsin π/4,
±ιcos π/4sin π/4
±sin π/4ιcos π/4,±±ιsin π/4ιcos π/4
ιcos π/4ιsin π/4,
±±sin π/4 cos π/4
cos π/4±sin π/4,±ιcos π/4ιsin π/4
ιsin π/4ιcos 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:C2R3
SO (3) R3SU (2)
C2
C2R3
φ
˜gAgψ(A)
φ
C2R3
,
˜gAgψ(A)C2R3
A SU (2) ψ(A)SO(3)
ψ:SU (2) SO (3)
gψ(A)(φ) = φgA).
(0, R)C2
r1, r2, r3R3(r1)2+ (r2)2+ (r3)2=R2RR
SU (2) SO (3)
C2R3z1, z2
p1+ιq1, p2+ιq2SU (2) ξ1, ξ 2, ξ3
SO (3) z1, z 2
SU (2) (0, R)C2z1, z 2
SO (3)
r1, r2, r3R3ξ1, ξ2, ξ3
z1, z2SU (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)2Im (z2)2(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
2Re z1z22Im z1z2z22z12
.
r1, r2, r3R3
1
R2
Re (z2)2(z1)2Im (z2)2(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
2Re z1z22Im z1z2z22z12
r1
r2
r3
.
R φ
SU (2) SO (3)
φ:z1
z27→ 1
R2
Re (z2)2(z1)2Im (z2)2(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
2Re z1z22Im z1z2z22z12
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={ASO(3), P (AX) = P(X), PV(AX) = PV(X), PD(AX) = P(X),XR3}.
C2
SU (2)
r1, r2, r3
φ:z1
z27→ 1
R2
Re (z2)2(z1)2Im (z2)2(z1)22Re z1z2
Re ι(z2)2+ (z1)2 Im ι(z2)2+ (z1)2 2Im z1z2
2Re z1z22Im z1z2z22z12
r1
r2
r3
z1, z2r1, r2, r3=a(1,ι, 0)
a
R3
z1, z2a=
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={USU (2), P (U Z ) = P(Z), PV(UZ) = PV(Z), PD(U Z) = P(Z),ZC2}.
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
Pz1, z2t=z1/z2
z2qP(q) z1
z2!=˜
z2qP(q) ˜
z1
˜
z2!,
q
P(8)
P(8)(t) = a8t8+a7t7+a6t6+a5t5+a4t4a5t3+a6t2a7t+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
2t2aV
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
2t2aD
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,eιπ/3z1)
±(e±ιπ/3z1, eιπ/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+a4t4a7t=t(a7t6+a4t3a7).
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, eιπ/4z2)
±(ιz1,ιz2)
±(ιz2, ιz1)
±(z2,z1)
±(ιe±ιπ/4z2,ιeιπ/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, eιθ/2z2)
±(eιθ/2z2,eιθ/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) = (tα)2qP(2q)(β
a
t+α
tα),
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, eιπ/4z2)
±(e±ιπ/4z2,eιπ/4z1)
±(cos(π/4)z1±ιsin(π/4)z2,±ιsin(π/4)z1+ cos(π/4)z2)
±(cos(π/4)z1sin(π/4)z2,±sin(π/4)z1+ cos(π/4)z2)
±(sin(π/4)z1+ιcos(π/4)z2, ι cos(π/4)z1sin(π/4)z2)
±(±ιsin(π/4)z1+ cos(π/4)z2,cos(π/4)z1ιsin(π/4)z2)
±(ιcos(π/4)z1sin(π/4)z2,±sin(π/4)z1ιcos(π/4)z2)
±(±ιsin(π/4)z1+ιcos(π/4)z2, ι cos(π/4)z1ιsin(π/4)z2)
±(±sin(π/4)z1+ cos(π/4)z2,cos(π/4)z1±sin(π/4)z2)
±(ιcos(π/4)z1ιsin(π/4)z2,ιsin(π/4)z1ιcos(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) 2q(tα)2qP(2q)(β
a
t+α
tα)
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=εq1a1
an=εqnan
a0=ωqa2q
a1=ω1qa2q1
an= (1)nωnqa2qn.
ε2= 1 q
ε2= 1 ω2= 1
a2k=a2q2k
ε3= 1 ω3= 1
ak
qk
ak= (1)kωkqa2qn
ε4= 1 ω4= 1
akqk
ak= (1)kωkqa2qn
|ε|= 1
n=q εnq= 1
ε4= 1
ω4= 1
α4= 1 β2= 1
akqk
ak= (1)kωkqa2qn
t= 1
α=1
an
n=q εnq= 1
SU (2)
... The simplest class is general anisotropy, while other symmetry classes (described as non-trivial) are the following: monoclinic symmetry, trigonal symmetry, orthotropic symmetry, tetragonal symmetry, transverse isotropy, cubic symmetry, and isotropy. The following symmetry groups correspond to the symmetry classes covered in this paper (Bona et al. 2004): ...
... Tensor c, representing a Hookean solid, belongs to one of eight material symmetry classes, as shown in several works (Forte & Vianello 1996, Bona et al. 2004. Symmetry classes are characterized by their symmetry groups, which are groups of transformations g (subgroups of 3D rotation group, SO(3)) leaving tensor c of given symmetry class invariant: ...
... Their positive definiteness is a favourable property to justify diffusion as a physical phenomenon. The conversion between 3D 4th order tensor coefficient and 6D 2nd order is obtained through equation (8). The factor 2 and √ 2 ensures isomorphism between the two spaces [26][32]. ...
Conference Paper
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Since its inception, DTI modality has become an essential tool in the clinical scenario. In principle, it is rooted in the emergence of symmetric positive definite (SPD) second-order tensors modelling the difusion. The inability of DTI to model regions of white matter with fibers crossing/merging leads to the emergence of higher order tensors. In this work, we compare various approaches how to use 4th order tensors to model such regions. There are three different projections of these 3D 4th order tensors to the 2nd order tensors of dimensions either three or six. Two of these projections are consistent in terms of preserving mean diffusivity and isometry. The images of all three projections are SPD, so they belong to a Rie-mannian symmetric space. Following previous work of the authors, we use the standard k-means segmentation method after dimension reduction with affinity matrix based on reasonable similarity measures, with the goal to compare the various projections to 2nd order tensors. We are using the natural affine and log-Euclidean (LogE) metrics. The segmentation of curved structures and fiber crossing regions is performed under the presence of several levels of Rician noise. The experiments provide evidence that 3D 2nd order reduction works much better than the 6D one, while diagonal components (DC) projections are able to reveal the maximum diffusion direction.
... Their positive definiteness is a favourable property to justify diffusion as a physical phenomenon. The conversion between 3D 4th order tensor coefficient and 6D 2nd order is obtained through equation (8). The factor 2 and √ 2 ensures isomorphism between the two spaces [26][32]. ...
... Taking into account this observation, it has been tried by some authors to use the harmonic factorization, according to Sylvester's theorem [29] and Maxwell's multipoles [30]. However, this involves roots' computations of polynomials of degree 4 and 8 [3,5,9], in order to build a set of 8 unit vectors (Maxwell's multipoles), without any clue of how to organize such data. Besides, Maxwell's multipoles are not, strictly speaking, first-order covariants of E and are moreover very sensitive to conditioning. ...
Article
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We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor. We produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants. An algorithm to detect the symmetry class of an Elasticity tensor is finally formulated.
... Taking account this observation, it has been tried by some authors to use the harmonic factorization, according to Sylvester's theorem [26] and Maxwell's multipoles [27]. However, this involves roots' computations of polynomials of degree 4 and 8 [4,5,8], in order to build a set of 8 unit vectors (Maxwell's multipoles), without any clue of how to organize such data. Besides, Maxwell's multipoles are not, strictly speaking, first-order covariant of E and are very sensitive to conditioning. ...
Preprint
We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor, once we know its symmetry class. In other words, we produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants.
... The object of this section is to explicitly describe this relationship, which is obtained using the Cartan map (see "Appendix B"). Although being rather overloaded in the field of continuum mechanics, this approach has been explored in some publications [8,9,19]. ...
Article
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We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity ten-sor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.
Article
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First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.
Preprint
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In the literature, there is an ambiguity in defining the relationship between trigonal and cubic symmetry classes of an elasticity tensor. We discuss the issue by examining the eigensystems and symmetry groups of trigonal and cubic tensors. Additionally, we present numerical examples indicating that the sole verification of the eigenvalues can lead to confusion in the identification of the elastic symmetry.
Article
Tensorial formulation of mechanical constitutive equations is a very important matter in continuum mechanics. For instance, the space of elastic tensors is a subspace of 4th order tensors with a natural SO(3) group action. More generaly, we have to study the geometry of a tensor space defined on R 3 , under O(3) group action. To describe such a geometry, we first have to exhibit its isotropy classes, also named symetry classes. Indeed, each tensor space possesses a finite number of isotropy classes. In this present work, we propose an original method to obtain isotropy classes of a given tensor space. As an illustration of this new method, we get for the first time the isotropy classes of a 8th order tensor space occuring in second strain-gradient elasticity theory. In the case of a real representation of a compact group, invariant algebra seperates the orbits. This observation motivates the purpose to find a finite generating set of poly- nomial invariants. For that purpose, we make use of the link between tensor spaces and spaces of binary forms, which belongs to the classical invariant theory. We thus have to deal with SL(2, C) group action. To obtain new results, we have reformulated and rein- terpreted effective approaches of Gordan’s algorithm, developped during XIXth century. We then obtain for the first time a minimal generating family of elasticity tensor space, and a generating family of piezoelectricity tensor space. Using linear algebra arguments, we were also able to get important relations of classical invariant theory, such as the Gordan’s series and the Abdesselam–Chipalkatti’s quadratic relations on transvectants.
Article
Tensorial formulation of mechanical constitutive equations is a very important matter in continuum mechanics. For instance, the space of elastic tensors is a subspace of 4th order tensors with a natural SO(3) group action. More generaly, we have to study the geometry of a tensor space defined on R 3 , under O(3) group action. To describe such a geometry, we first have to exhibit its isotropy classes, also named symetry classes. Indeed, each tensor space possesses a finite number of isotropy classes. In this present work, we propose an original method to obtain isotropy classes of a given tensor space. As an illustration of this new method, we get for the first time the isotropy classes of a 8th order tensor space occuring in second strain-gradient elasticity theory. In the case of a real representation of a compact group, invariant algebra seperates the orbits. This observation motivates the purpose to find a finite generating set of poly- nomial invariants. For that purpose, we make use of the link between tensor spaces and spaces of binary forms, which belongs to the classical invariant theory. We thus have to deal with SL(2, C) group action. To obtain new results, we have reformulated and rein- terpreted effective approaches of Gordan’s algorithm, developped during XIXth century. We then obtain for the first time a minimal generating family of elasticity tensor space, and a generating family of piezoelectricity tensor space. Using linear algebra arguments, we were also able to get important relations of classical invariant theory, such as the Gordan’s series and the Abdesselam–Chipalkatti’s quadratic relations on transvectants.
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Full-text available
The purpose of the present article is to give a precise definition and analysis from first principals of anisotropy, as the term applies to elastic media, taking care to avoid unnecessary assumptions. Two fundamental concepts, material invariance and symmetry group of a material, are defined purely in terms of the stress-strain relation. The implications of material symmetry, or in other words, of anisotropy, for the structure of the stiffness tensor are then investigated. Using the reduced notation of Voigt, these results are presented as the well-known simplifications in the form taken by the six-by-six stiffness matrix that represents the material's stiffness tensor. A new, simple proof is given for the remarkable fact that an elastic medium cannot have rotational symmetry by an angle of less than 90 without being transversely isotropic. In addition, the mutual relation that the notions of elastic symmetry and crystal symmetry have with respect to the so-called orthogonal group is sketched. Despite the historical association between anisotropic elastic materials and the study of crystals, the given presentation shows that conceptually the notion of anisotropy in elastic media is entirely independent of that of crystal symmetry.
Article
Two different definitions of symmetries for photoelasticity tensors are compared. Earlier for such symmetries the existence of exactly 12 classes was proved based on an equivalence relation induced on the set of subgroups of SO(3). Here, an another viewpoint is chosen, and photoelasticity tensors themselves are divided into symmetry classes, according to a different definition. By use of group-theoretical techniques, such as harmonic and Cartan decomposition, it is shown that this approach again leads to 12 classes.
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An insightful, structurally appealing and potentially utilitarian formulation of the anisotropic form of the linear Hooke's law due to Lord Kelvin was independently rediscovered by Rychlewski (1984, Prikl. Mat. Mekh.48, 303) and Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math.43, 14). The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor, considered as a second-rank tensor in six-dimensional space, are called eigentensors when projected back into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is therefore six. The concept of an eigentensor was introduced by Kelvin (1856, Phil. Trans. R. Soc.166, 481) who called eigentensors “the principal types of stress or of strain”. Kelvin determined the eigentensors for many elastic symmetries and gave a concise summary of his results in the 9th edition of the Encyclopaedia Britannica (1878). The eigentensors for a linear isotropic elastic material are familiar. They are the deviatoric second-rank tensor and a tensor proportional to the unit tensor, the spherical, hydrostatic or dilatational part of the tensor. Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math.43, 14) give explicit forms of the eigentensors for all of the linear elastic symmetries except monoclinic and triclinic symmetry. We discuss two approaches for the determination of eigentensors and illustrate these approaches by partially determining the eigentensors for monoclinic symmetry. With the nature of the eigentensors for monoclinic symmetry known, a rather complete table of the structural properties of all linear elastic symmetries can be constructed. The purpose of this communication is to give the most specifically detailed presentation of the eigenvalues and eigentensors of the Kelvin formulation to date.
Article
A complete and unified study of symmetries and anisotropies of classical and micropolar elasticity tensors is presented by virtue of a novel method based on a well-chosen complex vector basis and algebra of complex tensors. It is proved that every elasticity tensor has nothing but 1-fold, 2-fold, 3-fold, 4-fold and ∞-fold symmetry axes. From this fact it follows that the crystallographic symmetries plus the isotropic symmetry are complete in describing the symmetries of any kind of classical elasticity tensors and micropolar elasticity tensors. Further, it is proved that for each given integer m>>2 every classical Green elasticity tensor with an m-fold symmetry axis must have at least m elastic symmetry planes intersecting each other at this symmetry axis. From this fact and the aforementioned fact it follows that for all possible material symmetry groups, there exist only eight distinct symmetry classes for classical Green elasticity tensors, which correspond to the isotropy group and the seven crystal classes S 2, C 2h , D 2h , D 3d , D 4h , D 6h and O h , while it is shown that there exist twelve distinct symmetry classes for any other kind of elasticity tensors, including the classical Cauchy elasticity tensor and the micropolar elasticity tensors, which correspond to the eight subgroup classes just mentioned and the four crystal classes S 6, C 4h , C6h and T h . From these results, it turns out that all possible elasticity symmetry groups are nothing but the full orthogonal group, the transverse isotropy groups C ∞ h and D ∞ h , and the nine centrosymmetric crystallographic point groups except C 6h and D 6h .
Article
It is known from the theory of group representations that a general tensor can be expressed as a sum of traceless symmetric tensors. In this paper, based on Sylvester's theorem, it is shown that a general traceless symmetric tensor of any finite order m in three dimensions can be expressed as the traceless symmetric part of tensor product of m unit vectors (called the multipoles) multiplied by a positive scalar. The above two basic structures of tensors allow us to easily give complete and irreducible representations for tensor functions with high-order tensor variables, since those for tensor functions of vectors are well established in the literature. Examples are given for scalar-valued functions of a single fourth-order tensor of the elastic type, and of a number of vectors and second-order tensors. In 1970 Backus gave an alternative proof of Sylvester's theorem, which shows how to compute the multipoles. Since Backus's result is not so 'well known' to the community of researchers working on continuum mechanics, in the present paper a direct (without using Sylvester's theorem) and constructive establishment of Maxwell's multipole representation is provided, which is closer in spirit to a more modern approach to this topic.
Article
A geometrical picture of fourth-order, three-dimensional elastic tensors in terms of Maxwell multipoles is developed and used to obtain the elastic tensors appropriate to various crystal symmetry groups. Simply examining the picture shows whether an elastic tensor, described by its 21 independent components relative to an ill-chosen coordinate system, has an axis of symmetry of any order. The picture also facilitates obtaining the elastic tensor from the observed dependence of the three body-wave phase velocities on the direction of the propagation vector κ. In particular, for q = 2, 4, and 6, is a linear combination of the surface spherical harmonics of even orders up to and including q. Since determine uniquely, all the body-wave phase-velocity dependence on can be summarized by the 6 coefficients of spherical harmonics in P(2), the 15 coefficients in P(4), and the 28 coefficients in P(6). For nearly isotropic media, the anisotropy in the P velocity determines 15 of the 21 elastic coefficients, whereas determines the other 6 elastic coefficients. Our description of elastic tensors is generalized to all fourth-order tensors in three dimensions and certain fourth-order tensors in higher dimensions. The problem in higher dimensions produces simple examples of unitary representations of the rotation group ON+ with N ≥ 4 which contain no harmonic irreducible components.
Article
The Cowin–Mehrabadi theorem is generalized to allow less restrictive and more flexible conditions for locating a symmetry plane in an anisotropic elastic material. The generalized theorems are then employed to prove that the number of linear elastic symmetries is eight. The proof starts by imposing a symmetry plane to a triclinic material and, after new elastic symmetries are found, another symmetry plane is imposed. This process exhausts all possibility of elastic symmetries, and shows that there are only eight elastic symmetries. At each stage when a new symmetry plane is added, explicit results are obtained for the locations of the new symmetry plane that lead to a new elastic symmetry. It takes as few as three, and at most five, symmetry planes to reduce a triclinic material (which has no symmetry plane) to an isotropic material for which any plane is a symmetry plane.
Article
It is shown here that there are exactly eight different sets of symmetry planes that are admissible for an elasticity tensor. Each set can be seen as the generator of an associated group characterizing one of the traditional symmetry classes.
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It is shown that all symmetries possible for the elastic tensors can be reduced to the twelve symmetries already used in the description of the crystal classes. Each symmetry can be characterized by a group of rotations generated by no more than two rotations. The use of a canonical basis related to such rotations considerably simplifies the component forms of the elasticity tensor. This result applies to non-symmetric tensors; for symmetric tensors, the number of independent symmetries reduces from twelve to ten. After the present work was submitted, the following paper came to our attention: 14. S.C. Cowin and M.M. Mehrabadi, On the identification of material symmetry for anisotropic elastic materials. Q. Jl. Mech. appl. Math.40 (1987) 451–476. This paper contains an independent analysis of the partial ordering ≺ among the crystallographic elastic symmetries. However, it does not deal with the problem of the completeness of these symmetries.
Article
Harmonic and Cartan decompositions are used to prove that there are eight symmetry classes of elasticity tensors. Recent results in apparent contradiction with this conclusion are discussed in a short history of the problem.