Page 125 - Introduction to Continuum Mechanics

110 Kinematics of a Continuum

2Z>23 = rate of decrease of angle (from —) of two elements in 62 and 63 directions.

These rates of decrease of angle are also known as the rates of shear, or shearings.
Also, the first scalar invariant of the rate of deformation tensor D gives the rate of change
of volume per unit volume (see also Prob. 3.46). That is,

Or, in terms of the velocity components, we have

Since D is symmetric, we also have the result that there always exists three mutually
perpendicular directions (eigenvectors of D) along which the stretchings (eigenvalues of D)
include a maximum and a minimum value among all differential elements extending from a
material point.

Example 3.13.1

Given the velocity field:

(a) Find the rate of deformation and spin tensor.
(b) Determine the rate of extension of the material elements:

(c) Find the maximum and minimum rates of extension.
Solution, (a) The matrix of the velocity gradient is

so that