Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 334

The last term in this decomposition is still the skew-symmetric part of the
matrix. The second term is the traceless symmetric part of the matrix and the first
term is simply the trace of the matrix multiplied by the unit matrix.

Example A.3.1
Construct the three-way decomposition of the matrix A given by:

2 3

123
456
789

4 5 :

A ¼

Solution: The symmetric and skew-symmetric parts of A , A S , and A A , as well as the
trace of A are calculated,

2
4

3
5;

2
4

3
5;

135
357
579

0 1 2

1
2 ð

1
2 ð

A S A T A A A T ¼ A þ Þ¼ ¼ A Þ¼ 10 1

210

tr A ¼ 15 ;

then, since n ¼ 3, it follows from (A.16) that

2 3 2 3 2 3

500
050
005

435
307
574

0 1 2

4 5 þ 4 5 þ 4 5 :

A ¼ 10 1

210

Introducing the notation for the deviatoric part of an n by n square matrix A ,

tr A
n

dev A ¼ A 1 ; (A.17)

the representation for the matrix A given by (A.16) may be rewritten as the sum of
three terms

tr A
n

dev A S A A A ¼ 1 þ þ ; (A.18)

where the first term is called the isotropic ,or spherical (or hydrostatic as in
hydraulics) part of A . Note that

1
2 ð

dev A S dev A T ¼ dev A þ Þ: (A.19)

Example A.3.2

Show that tr (dev A ) ¼ 0.

Solution: Applying the trace operation to both sides of (A.17) one obtains