Page 493 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 477

Similarly, for all T

Since x^ = X2 for all r, therefore, from Eq. (ii)

Thus

from which

and

Thus,

8.6 Relative Deformation Gradient
1
Let dx and dx be the differential vectors representing the same material element at time
t and r, respectively. Then they are related by

That is

The tensor

is known as the relative deformation gradient. Here, the adjective "relative " indicates that
the deformation gradient is relative to the configuration at the current time. We note that for
r = t, dx' = dx so that

In rectangular Cartesian coordinates,