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

• Write a Matlab programme 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 invariants. 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 invariants 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.