Page 162 - Introduction to Continuum Mechanics

Kinematics of a Continuum 147

emanating from X.. The rectangular volume formed by cDP \ (DP ' and cflP ' at the reference
time t 0 is given by

(1) (l} (1 (2) (2) (2)
At time t, dX deforms into da = ¥dX \ rfX deforms into <& = F^X and
(3) (3)
dX^deforms into dx = FrfX and the volume is

That is,

T r
Since C = F F and B = FF , therefore

Thus, Eq. (3.29.3) can also be written as

We note that for an incompressible material, dV = dV 0 , so that

We note also that due to Eq. (3.29.3), the conservation of mass equation can be written as:

Example 3.29.1
Consider the deformation given by

(a)Find the deformed volume of the unit cube shown in Fig. 3.14.
(b)Find the deformed area of OABC.
(c) Find the rotation tensor and the axial vector of the antisymmetric part of the rotation tensor.
Solution, (a) From Eq. (i),