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A Primer in (Continuum) Mechanics
Homework assignment
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Draw a square and a parallelogram.

Describe the affine deformation from the first shape (the reference shape) to the second one (the current shape).
Compute the polar decomposition of the deformation gradient F.
Illustrate the corresponding rotation and stretch by a drawing.

Looking at the square as a face of a cube, extend the 2D deformation to a 3D deformation, 
just by adding a third row and a third column to the matrix of F, 
with only the (3,3) entry different from zero.
The cube will be transformed into a parallelpiped by the extended deformation.
Compute the ratio between the current shape volume and the reference shape volume.

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Optional extension.

Put the current shape in motion by choosing a periodic time law for the principal stretches
and a linear time law for the rotation amplitude. 
Compute the resulting time dependent deformation gradient.
Compute the corresponding velocity gradient and its decomposition into a symmetric and a skew-symmetric part 
(i.e. the stretching and the spin).
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