Page 52 - Introduction to Continuum Mechanics

PartB The Dual Vector of an Antisymmetric Tensor 37

Example 2B 16.1
Given
"l 2 3
[T] = 42 1
1 1 1
(a)Decompose the tensor into a symmetric and an antisymmetric part.
(b)Find the dual vector for the antisymmetric part.

(c)Verify T^a = ^xa for a = ^+e 3.
4
Solution, (a) [T] = [T^fT ], where
^ m±[lf [32 i
=
] =
2
[21 1

T 0 —1 1
[T4 ] = [T]-[T] = 1 Q Q
2
[-10 0
(b)The dual vector of i is
4
f = -(72^+^2+^3) = -(Oe!-e 2-e 3) = e 2+e 3.
(c) Let b = T^a, then
0 -1 l] [Y| I" l"
[b]= 1 000 = 1
[-1 0 OJ [ij [-1
i.e.,

b = e!+e 2-e 3
On the other hand,

t^xa = (e2+e 3)x(e 14-e 3) = -e 3+e!+e 2 = b

Example 2B 16.2

Given that R is a rotation tensor and that m is a unit vector in the direction of the axis of
rotation, prove that the dual vector q of K is parallel to m.

Solution. Since m is parallel to the axis of rotation, therefore,