Tensors for Physics

70 5 Symmetric Second Rank Tensors

I 2 =

2

3

s^2 + 2 q^2 , I 3 =3 det

(

S

)

= 2 s

(

1

9

s^2 −q^2

)

. (5.45)

The special planar biaxial case considered in (5.14) meanss=0, then one has
I 3 =0. On the other hand,I 3 is also zero forq=±^13 s. In this case the tensor is
also planar biaxial, but the roles of the principal axes 1, 2 ,3 are interchanged. For
a uniaxial tensor corresponding toq=0, the ratioI 32 /I 23 is equal to 1/6. Thus a
biaxiality parameterbcan be defined by

b^2 = 1 − 6 I 32 /I 23. (5.46)

Clearly, one hasb^2 =1 for the planar biaxial case corresponding tos=0 and
b^2 =0 for a uniaxial tensor withq=0.
For the dyadic tensor 2uμvνconstructed from the components of two unit vectors
uandv, as considered in Sect.5.2.5, one hass=−candq=1, withc:=u·v=
cosθ, whereθis the angle between the two vectors. Thenb^2 is equal to

1 − 3 c^2

(

1 −c^2 / 9

) 2 (

c^2 / 3 + 1

)− 3

.

In Fig.5.3, the biaxiality parameterbis plotted as function ofσ=sin^2 θ= 1 −c^2 ,
as given by

b=

[

1 −( 1 −σ)( 8 +σ)^2 ( 4 −σ)−^3

] 1 / 2

. (5.47)

Thiscurvedoesnotdeviatestronglyfromthedashedstraightlineb=σ.Asexpected,
the biaxiality parameter is zero forσ=0 corresponding toθ=0. It assumes its
maximum value 1 forσ=1 which pertains toθ=±π/2.

Fig. 5.3 The biaxiality
parameterbas function of
σ=sin^2 θ,whereθis the
angle between the vectorsu
andv, which form the
symmetric traceless dyadic.
The dashed line corresponds
tob=σ

0.2 0.4 0.6 0.8 1

σ

0.2

0.4

0.6

0.8

1

b