Page 160 - Introduction to Continuum Mechanics

Kinematics of a Continuum 145

We note that for infinitesimal deformation, Eq. (3.27.1) reduces to

which are the same as those given in Sect. 3.16.

3.28 Change of Area due to Deformation

Consider two material elements dfX^ = ^S^and^X^ - d$i*2 emanating from X.. The
rectangular area formed by dX^ and dX^ at the reference time t 0 is given by

where dA 0 is the magnitude of the undeformed area and 63 is normal to the area. At time
(2)
(1)
(1)
2
t, dX^ deforms into dx = FdX and dX^ deforms into dtf * = FdX and the area is
(1) (2)
dk = FdX x FdX = dSi dS 2 Fej xFe 2 = dA 0 Fe t X Fe 2 (3.28.2)
Thus, the orientation of the deformed area is normal to Fej and Fe^ Let this direction be
denoted by the unit vector n, i.e.,

then,

From the above equation, it is clear that

and