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    The Rotation Matrix

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    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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