Memo on the symmetry classes in elasticity
Triclinic
Monoclinic
(Reflection (e1 , e2 ))

•







•
•
•
•
•
•
•
•
•
•
•
•
•
•
S

•
•

•

•

•
•

•







•
•
•
•
•
0
0
0
•
S
⇒ 21(18) coefficients
0
0
0
•
•

•
•

•

0

0
•
⇒ 13(12) coefficients
Tetragonal
Orthotropic
(Orthotropic + directions e1 = e2 )

•







•
•
•
•
•
0
0
0
•
0
0
0
0
•
S

0
0

0

0

0
•

•¯







(Rotation of 2π/3 about e3
+ Reflection (e1 , e3 ))
•ˆ
•ˆ
•

0
0 


0
√ 
− 2´•

0 
?
•´
−´•
0
0
•¯
0
0
0
•¯
S

•¯







S
•
•¯
•ˆ
•ˆ
•
0
0
0
•¯
S
0
0
0
0
•¯

0
0

0

0

0
?
⇒ 5 coefficients
Isotropic
(Orthotropic + directions e1 = e2 = e3 )
•ˆ
•ˆ
•¯

0
0

0

0

0
•
(Isotropy in the plane (e1 , e2 ))
Cubic
•ˆ
•¯
0
0
0
0
•¯
Transversely isotropic
⇒ 6 coefficients

•¯







0
0
0
•¯
⇒ 6 coefficients
Trigonal
•
•¯
•ˆ
•ˆ
•
S
⇒ 9 coefficients

•¯







•
•¯
0
0
0
•¯
0
0
0
0
•¯
⇒ 3 coefficients

0
0

0

0

0
•¯
(Cubic + isotropy in the plane (e1 , e2 ))








•¯
•ˆ
•¯
•ˆ
•ˆ
•¯
S
0
0
0
?
0
0
0
0
?

0
0

0

0

0
?
⇒ 2 coefficients
Notation : • indicates independent coefficients. The other symbols (¯•, •ˆ, etc.) denote equal
components and ? = •1111 − •1122 .