Tensors for Physics

58 5 Symmetric Second Rank Tensors

5.2.4 Biaxial Tensors.

The general symmetric second rank tensor with three different principal values is
referred to asbiaxialtensor. With the abbreviations

S ̄=^1

3

(

S(^1 )+S(^2 )+S(^3 )

)

, s=S(^3 )−

1

2

(

S(^1 )+S(^2 )

)

, q=

1

2

(

S(^1 )−S(^2 )

)

,

(5.11)

the decomposition of the tensor according to (5.1), can be written as

Sμν=S ̄δμν+se(μ^3 )e(ν^3 )+q

(

e(μ^1 )eν(^1 )−e(μ^2 )e(ν^2 )

)

. (5.12)

To check the validity of this relation, notice thatS(i)=e(μi)Sμνe(νi)ande(μi)e
(j)
μe

(j)
ν e
(i)
μ
is equal to 2/3, fori=jand equal to− 1 /3, fori=j,cf.(3.27) and (3.28). Both,
iandj, can be equal to 1,2 or 3. The result is

S(^1 )=S ̄−

1

3

s+q, S(^2 )=S ̄−

1

3

s−q, S(^3 )=S ̄+

2

3

s. (5.13)

Furthermore, notice thateμ(^1 )e(ν^1 )−e(μ^2 )e(ν^2 )=e(μ^1 )e(ν^1 )−eμ(^2 )e(ν^2 ). Alternatively, to
obtain (5.12), one may start from (5.4) and decompose the tensor into its isotropic
and anisotropic parts according to

S=

1

3

∑^3

i= 1

S(i)δμν+

∑^3

i= 1

S(i)e(μi)e(νi).

Since

S(^1 )eμ(^1 )e(ν^1 )+S(^2 )e(μ^2 )e(ν^2 )

is equal to

1

2

(

S(^1 )+S(^2 )

)(

eμ(^1 )e(ν^1 )+e(μ^2 )e(ν^2 )

)

+

1

2

(

S(^1 )−S(^2 )

)(

eμ(^1 )e(ν^1 )−e(μ^2 )e(ν^2 )

)

and

e(μ^1 )eν(^1 )+e(μ^2 )e(ν^2 )=−e(μ^3 )e(ν^3 ),

equation (5.12) is recovered.