Page 496 - Introduction to Continuum Mechanics

480 Calculations of the Relative Deformation Tensor

Thus,

Also, since

therefore,

Let dx = dsei be a material element at the current time t and dx! = ds'n be the same
material element at time r, (where ej is a unit vector in a coordinate direction and n is a unit
vector), then Eq. (8.7.8) gives

On the other hand, if dx' = ds'*i is a material element at time t and dx = dsn is the same
material element at current time t, then Eq. (8.7.9) gives

The meaning of the other components can be obtained using Eq. (8.7.8) and (8.7.9). [See also
Sections 3.23 to 3.26 on finite deformation tensors in Chapter 3. However, care must be taken
in comparing equations in those sections with those in this chapter because of the difference
in reference configurations.
We note that

8.8 Calculations of the Relative Deformation Tensor

(A)Rectangular Coordinates
With the motion given by:

Equations (8.7.5) and (8.6.4) give