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View Full Version : Re: R: Homogeneous transform? What one? For sure not that one

jweiss55
07-14-2005, 11:26 PM
Hi Paolo,

Paolo de Leva wrote:

> Luckily, the "mechanical definition" is coherent with the etymology
>of the word "homogeneous". However, unfortunately I believe that, contrary
>to what Jeffrey wrote, his "mechanical definition" is not applicable to
>shearing. Shearing does not seem to be an "homogeneous transformation",
>neither according to the "math definition" nor according to the "mechanical
>definition". In fact, with shearing, the transformation along the x axis
>depends on the y and/or z position of the point, i.e. the x coordinate is
>multiplied by a coeefficient s1*y+s2*z, which is clearly dependent on the
>point position. Unfortunately, shearing is one of the affine transformations
>typically performed by using the multi-purpose 4x4 general T.M. we have been
>
>
The components of the deformation gradient itself are constant in space
(independent of material coordinates X, Y and Z) for simple shear. You
are confusing the deformation map with the deformation gradient. The
deformation gradient, which gives rise to the 3x3 rotation matrix used
in rigid body computations when it is proper orthogonal, comes from the
deformation map via differentiation. Thus the coordinate dependence is
differentiated away in the case of homogeneous deforrmations such as
simple shear.

This write-up from Rebecca Brannon should help:
Please see pages 14-16.

Here are some slides from my biomechanics class containing the widely
accepted "mechanical" definition of homogeneous deformation:

Here are a few slides from my class notes (based on Spencer's "Continuum
Mechanics" textbook) - they show how the deformation gradient gives rise
to stretching and rotation through the polar decomposition. See slides
2-10.