When developing constitutive models, it is crucial to run the model under a variety of standard (and some nonstandard) homogeneous deformations. To do this, you must first describe the motion mathematically. As indicated in , a good way to do that is to give the deformation gradient tensor, F. The component matrix [F] contains the deformed edge vectors of an initially unit cube, making this a very easy to way to prescribe deformations.
This posting links to a pdf, DelftExcerpts, which contains slides taken from a 2004 short course given in TU Delft (Netherlands) on the mathematics of tensor analysis. Following a review of the mathematics of line integrals, inexact differentials, and integrability, this set of slides provides some insight into the distinction between a global basis change (equivalent to the “space rotation” in the slides) and superimposed rotation. It also provides an introduction to the principle of material frame indifference (PMFI) as it applies to restricting allowable forms and input/output variables of computational constitutive models.