Page 504 - Introduction to Continuum Mechanics

488 Calculations of the Relative Deformation Tensor

where k ~ dv/dx-i-

Example 8.9.2

Given an axisymmetric velocity field in cylindrical coordinates:

(a) Obtain the motion using current time t as reference
(b) Compute the relative deformation tensor Q
(c) Find the Rivlin-Ericksen tensors.
Solution, (a) Let the motion be

then, from the given velocity field, we have

Integration of these equations with the conditions that at r = /, r'=r, 6' = 6 and 2' = 2, we
obtain

(b) Using Eq. (8.8.9), we obtain, with k(r) = dv /dr

(c)From the result of part (b), we have

Thus, the Rivlin-Ericksen tensors are [see Eq. (8.9.3)]