# OOF: Finite Element Analysis of Microstructures

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#

Rotations

The *orientation*variable in figure 4.1 illustrates how the orientation of the material in an element is specified. In the general case, three angles are required to specify an orientation. We use the Euler angles to describe one rotation as the result of three simpler rotations. Historically, Euler angles have been the source of much confusion, so, as our contribution to history, we present multiple ways of thinking about them.

*Different authors use different conventions, so don't assume that our definition is the same as yours!*

A rotation is entered as three values, in degrees, with square
brackets: `[ , , ]`. If only one
number is specified (without brackets), it is taken as the
value. That is, `30` is equivalent to `[0, 30, 0]` and is the
same as rotating about an axis normal to the *OOF* screen. Note that
`[0, 0, X]` is also equivalent to `[0, X, 0]`.

We define
to be the
screen coordinate system, with pointing to the right,
upwards, and out of the screen. We will
call the crystal axes
.
As usual, all rotations are right handed, *ie.* a positive
rotation about an axis is counterclockwise if the axis is pointing
towards you.

**One way of finding the Euler angles:**
This method describes the rotation that must be applied to the crystal
axes to bring them into alignment with the screen axes, assuming that
the crystal *starts* in its desired orientation with respect to
the screen. First, rotate the axes by around the
axis. This defines new axes
. Next rotate by about the axis, defining
another coordinate system
. Finally rotate by about the axis,
bringing the axes into agreement with
.

**An equivalent way:**
Consider the material oriented with a globe, with the origin at the
center of the earth and -axis pointing towards the north
pole, the -axis pointing at the Greenwich Meridian where
it intersects the equator and the -axis pointing towards
the Indian ocean somewhere southeast of Sri Lanka. The unrotated globe
has its , , and axes aligned
with the , , and axes of the
screen, respectively. The rotation triplet describes how to rotate
the *material* into its desired orientation. The first number of
the rotation triplet tilts the point on the north pole southward along
the Greenwich Meridian by degrees to a new latitude. The
second number spins the globe to the east by degrees about
its (tilted) -axis. The third number rotates the tilted
globe to the east by degrees about the screen's axis.

**Yet another way:**
The order of rotations in the previous definition can be rearranged.
Rotate the globe to the east by degrees about the -axis, tilt the north pole by about the -axis, and rotate to the east by again about the
-axis.

**A prescription:**
If you know the components in the crystal coordinate system of the
screen vectors , , and , you
can compute the Euler angles as follows. Let
be the
components of the screen's -axis in the crystal
coordinate system. Let
and
similarly be the components of the and -axes. Then

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**Previous:**Plane Stress and Plane

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