Tensors for Physics

56 5 Symmetric Second Rank Tensors

In solid state mechanics, the symmetric traceless part is commonly referred to
deviatoricpart,becauseitindicatesadeviationfromisotropy.Notice,notonlyvarious
names are used for the same item, also different notations are found in the literature.

5.2 Principal Values

5.2.1 Principal Axes Representation

A symmetric tensorScan be brought into a diagonal form with the help of an
appropriate rotation of the coordinate system. Then, in thisprincipal axes system,it
is presented as

S:=

⎛

⎝

S(^1 ) 00

0 S(^2 ) 0

00 S(^3 )

⎞

⎠, (5.3)

with realprincipal values, also calledeigenvalues S(i),i = 1 , 2 ,3. The axes of
the particular coordinate system in which the tensor is diagonal are referred to as
principal axes. Unit vectors parallel to these axes are denoted bye(i),i=1, 2, 3.
The order 1, 2, 3 can be chosen conveniently. With the help of the dyadicse(i)e(i),
the symmetric tensorScan be written in theeigen-representation:

S=

∑^3

i= 1

S(i)e(i)e(i). (5.4)

The validity of the standard eigenvalue equation

S·e(i)=S(i)e(i) (5.5)

follows from (5.4) and the orthogonalitye(i)·e(j)=δ(ij)of the eigen vectorse(i).
Notice that a symmetric second rank tensor in 3D has 6 independent components.
How come there are only 3 numbers specified by the 3 eigenvalues? The answer is:
3 additional numbers, e.g. Eulerian angles, are needed to determine the directions of
the principal axis in relation to an arbitrary, space fixed, coordinate system.

5.2.2 Isotropic Tensors

In the special case where all three principal values are equal,

S(^1 )=S(^2 )=S(^3 )=S,