Page 156 - The Combined Finite-Discrete Element Method

DEFORMATION GRADIENT 139

Any rotation in three-dimensional space is characterised by the axis ψ of rotation and the
angle of rotation ψ:
ψ =|ψ| (4.35)
For any vector a, the rotated configuration using this alternative way of calculating rotation
is given as follows:

1 1
Ra = (ψa)ψ + a − (ψa)ψ cos(ψ)+ (4.36)
ψ 2 ψ 2
1
(ψ × a) sin(ψ)
ψ

The matrix of rotation tensor with respect to the orthonormal basis
ψ

e 1 , e 2 , (4.37)
|ψ|
(where e 1 and e 2 are arbitrary unit vectors orthogonal to each other and orthogonal to
vector ψ)isgiven by
 
cos ψ − sin ψ 0
R =  sin ψ cos ψ 0  (4.38)
0 0 1
Physical interpretation of rotation is given in Figure 4.6. Material element does not change
its shape or size. It does not translate in space either. The material element rotates about
axis ψ instead

4.2.2 Transformation matrices

Base vectors of a deformed local frame can be expressed using base vectors of the
local frame:

˜
˜
i = i x i + i y j + i z k (4.39)
˜
˜
Deformed (rotated)
elemental volume

Initial elemental volume

ψ

Figure 4.6 Rotation about point q.