Diploma Thesis - Erich Schmid Institute

5. Orthogonal tensors and rotation
5.1. Basic relations
Orthogonal tensors and rotation
Crystals are anisotropic bodies, which means that the mechanical behaviour depends
on the crystal orientation. For this reason, it is important to exactly define the
orientation of the crystal in the sample coordinate system. Generally the sample and
the crystal coordinate systems can be related by a rotation (Chapter 4.2) expressed
by an orthogonal tensor O.
We consider two right angled coordinate systems with the unit vectors ei and ei’
(figure 5.1). A unit vector belonging to one coordinate system, for example ei, is
projected on the axes of the other system so that the position of this vector can be
obtained in the new coordinate system ei’ [7]. The length of the projected unit vector
on one axis is:
Figure 5.1: Relation between two tilted coordinate systems
e i ' . e j = ei
' e j cos( ei
',
e j)
e i ' = 1
j 1 = e
e . e = cos( e ',
e )
(equ. 5.1)
i ' j
i j
By applying equation 5.1 on all unit vectors of one coordinate system, an orthogonal
tensor is received that contains only cosine functions. Such a expression is also
known as direction cosine. For example let us use e1 to express e1‘, where the
components of e1‘ and e1 are x1’, y1’, z1’ and x1, y1, z1.
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