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The Rotation Matrix

The rotation matrix [R], associated with a positive (in a right-hand sense) rotation α ; about the z-axis is:

cosα sinα 0

-sinα cosα 0

0 0 1

? Derive the rotation Matrix [Q] relating (x, y, z) to a system (x', y', z') which is described by three consecutive Euler rotations about z, then y then x.

? Show that [Q] is orthogonal.

? The transformation rule for the strain tensor under a rotation is is [ε'] = [Q] [ε ] [Q]T . Why is not possible to find a matrix [A] such that [ε'] = [Q] [ε ] [Q]T Write down three quantities that you know to be invariant under the rotation.

? Write a Matlab program or construct an Excel spreadsheet to calculate the strain tensor under an arbitrary rotation in terms of given strains in a global (x,y,z) system, and to calculate the strain in-variants. Calculate the rotated strain tensor for Euler rotations of (45, 30, 30) degrees of the global strain tensor
[-0.1 0.05 0.0, 0.05 0.3 0.0, 0.0 0.0 -0.1]

? Calculate the strain in-variants in both systems, showing that they are same.
? Describe the deformed characteristics and sketch the deformation of a cube of material subjected to this strain.


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