Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 386

h i T

p T 23 ; p T 13 ; p T 12

T

T ¼½T 1 ; T 2 ; T 3 ; T 4 ; T 5 ; T 6

¼ T 11 ; T 22 ; T 33 ; ;

h i T

p J 23 ; p J 13 ; p J 12

T

J

¼½J 1 ; J 2 ; J 3 ; J 4 ; J 5 ; J 6 ¼ J 11 ; J 22 ; J 33 ; : (A.163)

These formulas permit the conversion of three-dimensional second order tensor
components directly to six-dimensional vector components and vice versa. The 2

factor that multiplies the last three components of the definition of the six-dimensional
vector representation of the three-dimensional second order tensor, (A.150), assures
the scalar product of the two six-dimensional vectors is equal to the trace of the
product of the corresponding second order tensors,

√

T

J ¼ T : J : (A.164)

The colon or double dot notation between the two second order tensors
illustrated in (A.164) is an extension of the single dot notation between the
matrices, A B , and indicates that one index from A and one index from B are to

be summed over; the double dot notation between the matrices, A : B , indicates that
both indices of A are to be summed with different indices from B . As the notation
above indicates, the effect is the same as the trace of the product, A : B

¼ tr( A B ).

A T : B in general. This
notation is applicable for square matrices in any dimensional space.
The vector U

A T : B T and A T : B A : B T but that A : B

Note that A : B ¼ ¼ 6¼

T is introduced to be the six-dimensional vector
representation of the three-dimensional unit tensor 1 . It is important to note that
the symbol U is distinct from the unit tensor in six dimensions that is denoted by 1.
Note that U

¼½ 1 ; 1 ; 1 ; 0 ; 0 ; 0

U 3, U T tr T and, using (A.164), it is easy to verify that T U ¼ ¼ ¼

tr T . The matrix C dotted with U yields a vector in six dimensions T : 1 ¼

2
4

3
5

c 11 þ c 12 þ c 13
c 21 þ c 22 þ c 23
c 31 þ c 32 þ c 33
^

C

U ¼ ; (A.165)

c 41 þ ^ c 42 þ ^ c 43

c 51 þ ^ ^ c 52 þ ^ c 53

c 61 þ ^ ^ c 62 þ ^ c 63

and, dotting again with U , a scalar is obtained:

U C U ¼ ^

c 11 þ ^ c 12 þ ^ c 13 þ ^ c 21 þ ^ c 22 þ ^ c 23 þ ^ c 31 þ ^ c 32 þ ^ c 33 : (A.166)

The transformations rules (A.161) and (A.162) for the vector and second order
tensors in six dimensions involve the six-dimensional orthogonal tensor transfor-
mation Q . The tensor components of Q are given in terms of Q by