Page 161 - Introduction to Continuum Mechanics

146 Change of Volume due to Deformation

Recall that for any vectors a, b, and c,
a-bxc = determinant whose rows are components of a, b, andc. Therefore

Eq. (iii) becomes

Using the definition of transpose of a tensor, Eqs. (ii) become

and Eq. (v) becomes

T
Thus, F n is in the direction of 63, so that

Therefore,

1 T*
Equation (3.28.4) states that the deformed area has a normal in the direction of (F ) 63 and
with a magnitude given by

In deriving Eq. (3.28.4), we have chosen the initial area to be the rectangular area formed by
the Cartesian base vectors ej and e 2, it can be shown that the formula remains valid for any
material area except that e 3 be replaced by the normal vector of the undeformed area n 0. That
is in general,

3.29 Change of Volume due to Deformation
Consider three material elements