5.5 Scalar Invariants of a Symmetric Tensor 69
5.5 Scalar Invariants of a Symmetric Tensor
5.5.1 Definitions.
The traceSμμ, the square of the magnitude (norm)SμνSμνand the determinant
det(S)=1
3
SμνSνλSλμ+1
6
(
SμμSνν− 3 SμνSνμ)
Sλλ (5.42)of the symmetric tensorSare not affected by a rotation of the coordinate system
and are therefore calledscalar invariants. More specifically, they are called scalar
invariants of first, second and third order, and denoted byI 1 ,I 2 ,I 3 , respectively. In
terms of the principal values, one has
Sμμ=S(^1 )+S(^2 )+S(^3 ), SμνSμν=S(^1 )
2
+S(^2 )
2
+S(^3 )
2
,and
det(S)=S(^1 )S(^2 )S(^3 ). (5.43)
To check the validity of (5.42), denote the principal values of the tensor bya,b,c
for simplicity. Then det(S)reads
1
3(
a^3 +b^3 +c^3)
+
1
6
(
(a+b+c)^2 − 3(
a^2 +b^2 +c^2))
(a+b+c)which is equal toabc, the result obtained directly from the determinant.
For symmetric traceless tensors, the symbolsI 2 andI 3 are used in the following,
for the norm and for the determinant, multiplied by 3:
I 2 = Sμν Sνμ, I 3 =3 det(
S
)
= Sμν Sνλ Sλμ. (5.44)Aderivationof(5.42)and(5.44),basedontheHamilton-Cayleytheorem,ispresented
in Sect.5.6.
The third order invariant can be used to decide whether a tensor is uniaxial or
biaxial without diagonalizing the tensor. A measure for biaxiality is introduced next.
5.5.2 Biaxiality of a Symmetric Traceless Tensor
As discussed above, a biaxial symmetric traceless tensor can be decomposed into a
uniaxial part and a planar biaxial part, cf. (5.12), characterized by coefficientssand
q. More specifically, the principal values are−^13 s+q,−^13 s−q, and^23 s,cf.(5.13).
Thus one has, according to (5.44),